# Planar Turán number for balanced double stars

Xin Xu School of Sciences, North China University of Technology, Beijing,
China. Qiang Zhou Academy of Mathematics and Systems Science, Chinese Academy
of Sciences, Beijing, China, and University of Chinese Academy of Sciences,
Beijing, China. Tong Li22footnotemark: 2 Guiying Yan22footnotemark: 2

###### Abstract

Planar Turán number, denoted by $\mathrm{ex}_{\mathcal{P}}(n,H)$, is the
maximum number of edges in an $n$-vertex planar graph which does not contain
$H$ as a subgraph. Ghosh, Győri, Paulos and Xiao initiated the topic of the
planar Turán number for double stars. For balanced double star, $S_{3,3}$ is
the only remaining graph need to be considered. In this paper, we give the
exact value of $\mathrm{ex}_{\mathcal{P}}(n,S_{3,3})$, forcing the planar
Turán number for all balanced double stars completely determined.
Keywords: Planar Turán number, Balanced double stars, Extremal graphs.

##  1 Introduction

All graphs considered in this paper are finite, undirected and simple. Let
$V(G)$, $v(G)$, $E(G)$, $e(G)$, $\delta(G)$ and $\Delta(G)$ denote the vertex
set, number of vertices, edge set, number of edges, minimum degree and maximum
degree of a graph $G$, respectively. For any subset $S\subset V(G)$, the
subgraph induced on $S$ is denoted by $G[S]$. We denote by $G\backslash S$ the
subgraph induced on $V(G)\backslash S$. If $S=\\{v\\}$, we simply write
$G\backslash v$. We use $e[S,T]$ to denote the number of edges between $S$ and
$T$, where $S$, $T$ are subsets of $V(G)$.

Let $H$ be a graph, a graph is called $H$-free if it does not contain $H$ as a
subgraph. One of the most classical problems in extremal graph theory,
nowadays so-called Turán-type problem is:

######  Problem 1.1 (Turán Problem).

What is the maximum number of edges in an $n$-vertex $H$-free graph $G$?

We use $\mathrm{ex}(n,H)$ to denote the maximum number of edges in an
$n$-vertex $H$-free graph. In 1941, Turán [24] gave the exact value of
$\mathrm{ex}(n,K_{r})$ and the extremal graph, where $K_{r}$ is a complete
graph with $r$ vertices. Later in 1946, Erdős and Stone [5] extended this
result by replacing $K_{r}$ by an arbitrary graph $H$ and showed that
$\mathrm{ex}(n,H)=(1-\frac{1}{\chi(H)-1}+o(1))\binom{n}{2}$, where $\chi(H)$
denotes the chromatic number of $H$. This is latter called the “fundamental
theorem of extremal graph theory” [1].

In 2016, Dowden [3] initiated the study of Turán-type problems when host
graphs are planar graphs:

######  Problem 1.2 (Planar Turán Problem).

What is the maximum number of edges in an $n$-vertex $H$-free planar graph
$G$?

We use $\mathrm{ex}_{\mathcal{P}}(n,H)$ to denote the maximum number of edges
in an $n$-vertex $H$-free planar graph. Dowden studied the planar Turán number
of $C_{4}$ and $C_{5}$, where $C_{k}$ is a cycle with $k$ vertices. Ghosh,
Győri, Martin, Paulos and Xiao [9] gave the exact value for $C_{6}$. Shi,
Walsh and Yu [22], Győri, Li and Zhou [12] gave the exact value for $C_{7}$.
The planar Turán number of $C_{k}$ is still unknown for $k\geq 8$. Cranston,
Lidický, Liu and Shantanam [2] first gave both lower and upper bound for
general cycles, Lan and Song [20] improved the lower bound. Recently, Shi,
Walsh and Yu [23] improved the upper bound, Győri, Varga and Zhu [14] gave a
new construction and improved the lower bound. Lan, Shi and Song [17] gave a
sufficient condition for graphs with planar Turán number $3n-6$. We refer the
interested readers to more results on paths, theta graphs and other graphs
[16, 18, 11, 15, 26, 6, 7, 21, 19, 4, 13].

######  Definition 1.1.

A double star $S_{m,l}$ is the graph obtained by taking an edge $xy$ and
joining $x$ with $m$ vertices, $y$ with $l$ vertices which are different from
the $m$ vertices.

Figure 1: The double star $S_{m,l}$.

In 2022, Ghosh, Győri, Paulos and Xiao [10] studied the planar Turán number
for $S_{2,2}$, $S_{2,3}$, $S_{2,4}$, $S_{2,5}$, $S_{3,3}$ and $S_{3,4}$.
Moreover, they gave the exact value for $S_{2,2}$ and $S_{2,3}$. Later, The
first author of this paper improved the upper bound for $S_{2,5}$ [25].

We say a double star is a balanced double star if $m=l$. For $m\geq 4$,
$\mathrm{ex}_{\mathcal{P}}(n,S_{m,m})=3n-6$ since a double wheel graph (a
graph with two non-adjacent vertices connecting to all vertices on the cycle
$C_{n-2}$) does not contain $S_{m,m}$ as a subgraph. Since $S_{1,1}$ is a path
on 4 vertices and every graph without a path on 4 vertices must be a planar
graph, the planar Turán number of $S_{1,1}$ is equal to its Turán number. By
the result of Faudree and Schelp [8],
$\mathrm{ex}_{\mathcal{P}}(n,S_{1,1})=\mathrm{ex}(n,S_{1,1})\leq n$ and the
equality holds for $3|n$. Ghosh, Győri, Paulos and Xiao [10] proved that
$\mathrm{ex}_{\mathcal{P}}(n,S_{2,2})=2n-4$ when $n\geq 16$ and $\lfloor
5n/2\rfloor-5\leq\mathrm{ex}_{\mathcal{P}}(n,S_{3,3})\leq\lfloor
5n/2\rfloor-2$ when $n\geq 3$. Moreover, they gave the following conjecture:

######  Conjecture 1.1.

$\displaystyle\begin{split}\mathrm{ex}_{\mathcal{P}}(n,S_{3,3})=\left\\{\begin{array}[]{ll}3n-6&\text{if
}3\leq n\leq 7,\\\ 16&\text{if }n=8,\\\ 18&\text{if }n=9,\\\ \lfloor
5n/2\rfloor-5&\text{otherwise}.\end{array}\right.\end{split}$

In this paper, we solve this conjecture by a new method and thus the planar
Turán number for all balanced double stars is determined:

######  Theorem 1.1.

For any integer $n\geq 3$, we have

$\displaystyle\begin{split}\mathrm{ex}_{\mathcal{P}}(n,S_{m,m})=\left\\{\begin{array}[]{ll}n&\text{if
}m=1\text{ and }3|n,\\\ n-1&\text{if }m=1\text{ and }3\nmid n,\\\
2n-4&\text{if }m=2\text{ and }n\geq 16,\\\ \lfloor 5n/2\rfloor-5&\text{if
}m=3\text{ and }n\geq 10,\\\ 3n-6&\text{if }m\geq
4.\end{array}\right.\end{split}$

##  2 Definitions and Preliminaries

Let $G$ be an $S_{3,3}$-free planar graph. For the sake of brevity and
simplicity, we always omit the floor function.

######  Definition 2.1.

A $\boldsymbol{k\text{-}l}$ edge is an edge whose end vertices are of degree
$k$ and $l$. A $\boldsymbol{k\text{-}l\text{-}s}$ path is an induced path
consisting of three vertices with degree $k$, $l$ and $s$.

######  Definition 2.2.

A $\boldsymbol{k\text{-}s}$ star is a subgraph in $G$ on $k+1$ vertices, where
there is a central vertex connecting to the other $k$ vertices, and all other
$k$ vertices have degree of $s$. We use $\boldsymbol{k^{+}\text{-}s^{-}}$ star
to denote the one with the central vertex of degree at least $k$ and the other
$k$ vertices of degree at most $s$.

######  Definition 2.3.

Given two subgraphs $H,H^{\prime}\subset G$, we use
$\boldsymbol{H+H^{\prime}}$ to denote the subgraph induced on $V(H)\cup
V(H^{\prime})$. If $V(H^{\prime})=\\{v\\}$, we abbreviate this as $H+v$.

######  Definition 2.4.

The elementary star-block B of $G$ is one of $5^{+}$-$3^{-}$ star, $5$-$4^{-}$
star, $6$-$6$ edge, $6$-$5$ edge, $6$-$4$ edge, $5$-$5$ edge, $5$-$4$-$5$ path
in $G$. Given an elementary star-block B, a vertex $v\in V(G)\backslash V(B)$
is called potential if $N(v)\cap V(B)\neq\emptyset$ when $d(v)\leq 3$ or
$|N(v)\cap V(B)|=2$ when $d(v)=4$. If $B$ is an elementary star-block and
$V_{p}$ is a set of potential vertices, then $B$ and $B+V_{p}$ are both called
star-blocks. For any star-block $B$, the vertices in $\\{v\in V(B)|N(v)\cap
V(G\backslash B)\neq\emptyset\text{\ for\ }d(v)\leq 3\text{\ and\ }|N(v)\cap
V(G\backslash B)|=2\text{\ for\ }d(v)=4\\}$ are peripheral vertices.

Note that if a graph has a $k$-$l$ edge with $k\geq 7$ and $l\geq 4$, then $G$
contains an $S_{3,3}$ as a subgraph. So all vertices of degree at least 5 must
lie in some star-block defined above. Next we ensure that for any vertex $v$
with $d(v)\geq 5$, there is exactly one star-block containing $v$. The star-
block is the first one by checking in the order of $7^{+}$-$3^{-}$ star,
$6$-$6$ edge, $6$-$5$ edge, $6$-$4$ edge, $6$-$3^{-}$ star, $5$-$5$ edge,
$5$-$4$-$5$ path, $5$-$4^{-}$ star.

######  Definition 2.5.

Let $H$ be a subgraph of $G$. The star-block base $\mathcal{B}$ of $H$ is the
set consisting of star-blocks satisfying:
$(a)$ $V(H)=\bigcup\limits_{B\in\mathcal{B}}V(B)$;
$(b)$ $\forall B,B^{\prime}\in\mathcal{B}$, if $V(B)\cap
V(B^{\prime})\neq\emptyset$, then all common vertices are peripheral vertices
in both $B$ and $B^{\prime}$. Moreover, the common peripheral vertices are
called shared vertices.

It should be noticed that for any shared vertex $v$, we have $2\leq d(v)\leq
4$. In fact, if $d(v)=1$, $v$ belongs to one star-block and can not be shared.
If $v$ is a shared vertex with $d(v)\geq 5$, then an $S_{3,3}$ is easily
found.

######  Definition 2.6.

Let $G=G_{1}+G_{2}$. If $\mathcal{B}$ is a star-block base of $G_{1}$ and any
vertex in $V(G_{2})$ has degree at most $4$ in $G$, then we say $G$ has a
star-block partition. Let $d_{\mathcal{B}}(v)$ be the number of star-blocks in
base $\mathcal{B}$ containing $v$. For any star-block $B\in\mathcal{B}$, let
$d_{s}(B)$ denote the number of shared vertices of degree at most $3$ in $B$
and let $d_{s^{\prime}}(B)$ denote the number of shared vertices of degree $4$
in $B$.

It is easy to see that $G$ must have a star-block partition. Specially, if
$\Delta(G)\leq 4$, then $G=G_{2}$.

######  Definition 2.7.

Let $H$ be a subgraph of $G$. The primary weight of $H$, denoted by
$\boldsymbol{w_{0}(H)}$, is defined as

$\boldsymbol{w_{0}(H)}\coloneqq e(H)+\frac{1}{2}(e[H,G\backslash
H])=\frac{1}{2}\sum_{v\in V(H)}d(v).$

######  Definition 2.8.

Let $G$ have a star-block partition and a star-block base $\mathcal{B}$. For
any $B\in\mathcal{B}$, the modified weight of $B$, denoted by
$\boldsymbol{w(B)}$, is defined as

$\boldsymbol{w(B)}\coloneqq
w_{0}(B)+\frac{s}{2}+\frac{s^{\prime}}{4}+\mathbf{1}_{B},$

where $s=d_{s}(B)$, $s^{\prime}=d_{s^{\prime}}(B)$, and

$\displaystyle\begin{split}\mathbf{1}_{B}=\left\\{\begin{array}[]{ll}1&\text{if
}d_{\mathcal{B}}(v)=3\text{ for some }v\in B,\\\ 0&\text{otherwise}.\\\
\end{array}\right.\end{split}$

Obviously, if there is a star-block partition $G=G_{1}+G_{2}$, then
$e(G)=w_{0}(G)=w_{0}(G_{1})+w_{0}(G_{2})$.

######  Definition 2.9.

Let $G$ have a star-block base $\mathcal{B}=\\{B_{i},i=1,2,\cdots,T\\}$ and
$\mathcal{B}^{\prime}$ be the star-block base obtained from $\mathcal{B}$
satisfying: $(a)$ $|\mathcal{B}^{\prime}|=|\mathcal{B}|$; $(b)$ for all $1\leq
i\leq T$, there exist $B_{i}\in\mathcal{B}$ and
$B^{\prime}_{i}\in\mathcal{B}^{\prime}$ such that $V(B_{i})\subset
V(B^{\prime}_{i})$ and $w(B^{\prime}_{i})/v(B^{\prime}_{i})\leq
w(B_{i})/v(B_{i})$, where at least one inequality is strict. Then we say
$\mathcal{B}^{\prime}$ is the refinement of $\mathcal{B}$.

For the sake of convenience in subsequent discussion, we categorize the star-
blocks into three types:

  * $\bullet$

$\mathcal{B}_{0}\coloneqq\\{B\in\mathcal{B}\ |\
d_{s}(B)+d_{s^{\prime}}(B)=0\\}$,

  * $\bullet$

$\mathcal{B}_{1}\coloneqq\\{B\in\mathcal{B}\ |\ d_{s}(B)+d_{s^{\prime}}(B)\geq
1\ and\ d_{\mathcal{B}}(v)\leq 2\ for\ each\ v\in B\\}$,

  * $\bullet$

$\mathcal{B}_{2}\coloneqq\\{B\in\mathcal{B}\ |\ d_{s}(B)+d_{s^{\prime}}(B)\geq
1\ and\ d_{\mathcal{B}}(v)=3\ for\ some\ v\in B\\}$.

We show that $w(B)$ can be constrained for each star-block in some base
$\mathcal{B}$.

######  Lemma 2.1.

Let $G$ be an $S_{3,3}$-free planar graph on $n$ vertices. Then there exists a
star-block partition $G=G_{1}+G_{2}$ such that all star-blocks in the base
$\mathcal{B}$ satisfying:

$\displaystyle\begin{split}w(B)\leq\left\\{\begin{array}[]{ll}\frac{5}{2}v(B)-\frac{5}{2}&\text{when
}B\in\mathcal{B}_{0},\\\ \frac{5}{2}v(B)-\frac{5}{t}&\text{when
}B\in\mathcal{B}_{1},\\\ \frac{5}{2}v(B)-1&\text{when
}B\in\mathcal{B}_{2},\end{array}\right.\end{split}$

where $t=|\mathcal{B}_{1}|+|\mathcal{B}_{2}|$.

This lemma is the highlight of this paper, which make it possible to prove the
following two lemmas.

######  Lemma 2.2.

Let $G$ be an $S_{3,3}$-free planar graph on $n\geq 7$ vertices. If $G$
contains only one star-block, then

$\displaystyle\begin{split}e(G)\leq\left\\{\begin{array}[]{ll}15&\text{when
}n=7,\\\ 16&\text{when }n=8,\\\ 18&\text{when }n=9,\\\ 5n/2-5&\text{when
}n\geq 10.\end{array}\right.\end{split}$

######  Lemma 2.3.

Let $G$ be an $S_{3,3}$-free planar graph with a star-block partition
$G=G_{1}+G_{2}$ and a star-block base $\mathcal{B}$. If $|\mathcal{B}|\geq 2$,
then $e(G)\leq 5n/2-5$.

Together with the extremal graphs achieving the bound, Theorem 1.1 can be
deduced from these two lemmas, which will be proved in the following sections.

##  3 Proof of Lemma 2.1

Suppose that there exists a partition $G=G_{1}+G_{2}$ and a star-block base
$\mathcal{B}$. We show that if there exists a star-block $B\in\mathcal{B}$
which does not satisfy the corresponding upper bound, we can construct a
refinement $\mathcal{B}^{\prime}$ such that for some
$B^{\prime}\in\mathcal{B}^{\prime}$, $B\subset B^{\prime}$ and $w(B^{\prime})$
satisfies the bound.

There are several different star-blocks in $G$, such as $5^{+}$-$3^{-}$ star,
$6$-$6$ edge, $6$-$5$ edge, $6$-$4$ edge, $5$-$5$ edge, $5$-$4$-$5$ path,
$5$-$4^{-}$ star and their variant forms.

Now we consider each case in turn. Recall that $s$ the number of shared
vertices of degree at most $3$ and $s^{\prime}$ denote the number of shared
vertices of degree $4$ in $B$, $\mathbf{1}_{B}$ the characteristic function
for whether $B$ contains a $3$-degree vertex with $d_{\mathcal{B}}(v)=3$ or
not.

Case 1. $B$ is a $5^{+}$-$3^{-}$ star.

Assume that $B$ is a $k$-$3^{-}$ star for $k\geq 5$. We have

$\displaystyle w(B)$
$\displaystyle\leq\frac{1}{2}(k+3k)+\frac{s}{2}+\mathbf{1}_{B}$
$\displaystyle\leq\frac{5}{2}k+\mathbf{1}_{B}.$

If $\mathbf{1}_{B}=0$, then
$w(B)\leq\frac{5}{2}k=\frac{5}{2}(k+1)-\frac{5}{2}$. If $\mathbf{1}_{B}=1$,
then $w(B)\leq\frac{5}{2}k+1=\frac{5}{2}(k+1)-\frac{3}{2}<\frac{5}{2}(k+1)-1$.

Case 2. $B$ is a $6$-$6$ edge.

Let $uv$ be the $6$-$6$ edge. There exists at least $5$ triangles sitting on
the edge $uv$, otherwise an $S_{3,3}$ is found in $G$. Let
$a_{1},a_{2},a_{3}$, $a_{4}$ and $a_{5}$ be the vertices adjacent to both $u$
and $v$, as shown in Figure 2$(a)$. Let
$S_{1}=\\{a_{1},a_{2},a_{3},a_{4},a_{5}\\}$ and $H_{1}=G[S_{1}]$.

Figure 2: $(a)$ The star-block is a $6$-$6$ edge $uv$. $(b)$ The connected
component on $7$ vertices.

Note that vertices in $S_{1}$ can form a path of length at most $4$ and each
vertex in $S_{1}$ can have at most one neighbor in $V(G)\backslash B$,
otherwise $G$ contains an $S_{3,3}$. We have $e(H_{1})\leq 4$ and
$e[H_{1},G\backslash B]\leq 5$. Moreover, for any $x\in S_{1}$, if $d(x)=4$,
then $x$ can not have a neighbor outside of $B$, otherwise an $S_{3,3}$ is
also found. So the degree of shared vertices in $S_{1}$ is exactly $3$, which
implies $s^{\prime}=0$. Each shared vertex is shared by exactly two star-
blocks since it is adjacent to both $u$ and $v$, which means
$\mathbf{1}_{B}=0$.

Hence we have $w(B)=11+e(H_{1})+\frac{1}{2}e[H_{1},G\backslash
B]+\frac{s}{2}$.

Assume that $e(H_{1})=0$. Now we analyze the relationship between the number
of shared vertices in $B$ and the number of star-blocks in $G$. It can be
checked that

$\displaystyle\begin{split}t\geq\left\\{\begin{array}[]{ll}0&\text{when
}s=0,\\\ 2&\text{when }s=1,2,\\\ 3&\text{when }s=3,4,\\\ 4&\text{when
}s=5.\end{array}\right.\end{split}$

When $s=0$, $w(B)\leq 11+\frac{5}{2}=\frac{5}{2}\cdot 7-4<\frac{5}{2}\cdot
7-\frac{5}{2}$.

When $s=1,2$, $w(B)\leq 11+\frac{5}{2}+\frac{s}{2}\leq\frac{5}{2}\cdot
7-3<\frac{5}{2}\cdot 7-\frac{5}{t}$.

When $s=3,4$, $w(B)\leq 11+\frac{5}{2}+\frac{s}{2}\leq\frac{5}{2}\cdot
7-2<\frac{5}{2}\cdot 7-\frac{5}{t}$.

When $s=5$, $w(B)\leq 11+\frac{5}{2}+\frac{s}{2}\leq\frac{5}{2}\cdot
7-\frac{3}{2}<\frac{5}{2}\cdot 7-\frac{5}{t}$.

If $e(H_{1})=1$, assume that $a_{1}a_{2}\in E(G)$. Then $a_{1},a_{2}$ can not
have a neighbor outside and can not be shared either. This implies $s\leq 3$
and $e[H_{1},G\backslash B]\leq 3$. Then if $s=0$, we have $w(B)\leq
12+\frac{3}{2}=\frac{5}{2}\cdot 7-4<\frac{5}{2}\cdot 7-\frac{5}{2}$. If $s\geq
1$, then $w(B)\leq 12+\frac{3}{2}+\frac{s}{2}\leq\frac{5}{2}\cdot
7-\frac{5}{2}\leq\frac{5}{2}\cdot 7-\frac{5}{t}$.

Similarly, if $e(H_{1})=p$ for $2\leq p\leq 4$, we obtain that $s\leq 4-p$ and
$e[H_{1},G\backslash B]\leq 4-p$. It follows that $w(B)\leq
11+p+\frac{4-p}{2}+\frac{4-p}{2}=\frac{5}{2}\cdot 7-\frac{5}{2}$.

Specially, if $s=0$, $w(B)\leq 13+\frac{p}{2}$. When $p\leq 3$,
$w(B)\leq\frac{5}{2}\cdot 7-3$. When $p=4$, $B$ is a connected component in
$G$ with $e(B)=w(B)=15$, as shown in Figure 2$(b)$. Here, we list these two
results separately because they will play a role in the proof of the Lemma
2.2.

Case 3. $B$ is a $6$-$5$ edge.

Let $uv$ be the $6$-$5$ edge and $u$ be the vertex of degree 6. There exist at
least $4$ triangles sitting on the edge $uv$, otherwise an $S_{3,3}$ is found.
Let $a_{1},a_{2},a_{3}$ and $a_{4}$ be the vertices adjacent to both $u$ and
$v$, let $b_{1}$ be the vertex adjacent to only $u$, as shown in Figure
3$(a)$. Let $S_{1}=\\{a_{1},a_{2},a_{3},a_{4}\\}$, $S_{2}=\\{b_{1}\\}$ and
$H_{1}=G[S_{1}]$.

Figure 3: $(a)$ The star-block is a $6$-$5$ edge $uv$. $(b)$ The connected
component on $7$ vertices.

Similar to Case 2, each shared vertex in $S_{1}$ has degree exactly 3. We
claim that $d(b_{1})\leq 3$. In fact, if $d(b_{1})\geq 4$, there exists an
edge $b_{1}b^{\prime}_{1}\in E(G)$ and $b^{\prime}_{1}\notin S_{1}$, then $G$
contains an $S_{3,3}$.

Case 3.1. If $d_{\mathcal{B}}(b_{1})=3$, we have $\mathbf{1}_{B}=1$ and
$b_{1}$ is not adjacent to any vertex in $S_{1}$. Then
$w(B)=10+e(H_{1})+\frac{1}{2}e[B,G\backslash B]+\frac{s}{2}+1$. Assume that
$e(H_{1})=0$. It follows that $w(B)\leq
10+\frac{6}{2}+\frac{5}{2}+1=\frac{33}{2}=\frac{5}{2}\cdot 7-1$. If
$e(H_{1})\geq 1$, it is easy to verify that $w(B)$ will not increase, which
satisfies the upper bound too.

Case 3.2. If $d_{\mathcal{B}}(b_{1})=2$, it is obtained that
$\mathbf{1}_{B}=0$. Here we give the relationship between the number of shared
vertices in $B$ and the number of star-blocks in $G$. When $s=1,2,3$, $t\geq
2$. If $e(H_{1})=0$, we have $w(B)\leq
10+\frac{6}{2}+\frac{3}{2}=\frac{29}{2}=\frac{5}{2}\cdot 7-3$. If
$e(H_{1})\geq 1$, it is easy to check that $w(B)$ will not increase. When
$s=4,5$, it is easy to see that $t\geq 3$ and $e(H_{1})=0$. We have $w(B)\leq
10+\frac{6}{2}+\frac{5}{2}=\frac{31}{2}<\frac{5}{2}\cdot 7-\frac{5}{t}$.

Case 3.3. If $d_{\mathcal{B}}(b_{1})=1$, then $s\leq 4$.

When $s=0$, we discuss the subcases based on the degree of $b_{1}$ in $B$. If
$|N(b_{1})\cap S_{1}|=0$, the discussion here is essentially the same as the
subcase when $d_{\mathcal{B}}(b_{1})=3$, where $w(B)$ here is reduced by
exactly $\mathbf{1}_{B}+\frac{1}{2}$. So $w(B)\leq\frac{5}{2}\cdot
7-\frac{5}{2}$. If $|N(b_{1})\cap S_{1}|=1$, the proof is same as the subcase
when $d_{\mathcal{B}}(b_{1})=2$. And $w(B)$ is reduced by at least
$\frac{1}{2}$. Then $w(B)\leq\frac{31}{2}-\frac{1}{2}=\frac{5}{2}\cdot
7-\frac{5}{2}$. In fact, a more refined calculation can yield that
$w(B)\leq\frac{5}{2}\cdot 7-3$.

It remains to prove the subcase when $|N(b_{1})\cap S_{1}|=2$. Assume that
$b_{1}a_{1},b_{1}a_{4}\in E(G)$. Then $b_{1},a_{1},a_{4}$ all have no neighbor
outside of $B$, which implies $s\leq 2$. If $e(H_{1})=p$ for $1\leq p\leq 3$,
we obtain that $s\leq 3-p$ and $e[B,G\backslash B]=e[H_{1},G\backslash B]\leq
3-p$. Hence $w(B)\leq 12+p+\frac{3-p}{2}+\frac{3-p}{2}=\frac{5}{2}\cdot
7-\frac{5}{2}$. Specially, it can be checked that the equality holds if and
only if $B$ is a connected component, as shown in Figure 3$(b)$.

When $s=1,2$, we obtain $e(H_{1})\leq 2$ and $t\geq 2$. If $e(H_{1})=0$, it
follows that $w(B)\leq 10+\frac{6}{2}+\frac{2}{2}=\frac{5}{2}\cdot
7-\frac{7}{2}$. And if $e(H_{1})\geq 1$, it is easy to see that $w(B)$ does
not increase.

When $s=3,4$, $e(H_{1})=0$ and $t\geq 3$. We have that $w(B)\leq
10+\frac{6}{2}+\frac{4}{2}=\frac{5}{2}\cdot 7-\frac{5}{2}$.

Case 4. $B$ is a $6$-$4$ edge.

Let $uv$ be the $6$-$4$ edge, as shown in Figure 4. There exist at least $3$
triangles sitting on the edge $uv$, otherwise an $S_{3,3}$ is found. Let
$S_{1}=\\{a_{1},a_{2},a_{3}\\}$, $S_{2}=\\{b_{1},b_{2}\\}$ and
$H_{1}=G[S_{1}]$.

Figure 4: The star-block $B$ is a $6$-$4$ edge $uv$.

Case 4.1. $d_{\mathcal{B}}(b_{1})=3,d_{\mathcal{B}}(b_{2})\leq 3$.

Note that $b_{1}$ is not adjacent to any vertex in $S_{1}\cup\\{b_{2}\\}$.
Each vertex in $\\{b_{1},b_{2}\\}$ has at most two neighbors outside.

If $e(H_{1})=0$ and $e[b_{2},S_{1}]=0$, then $w(B)\leq
9+\frac{7}{2}+\frac{5}{2}+1=16<\frac{5}{2}\cdot 7-1$. It is easy to check that
the conditions $e(H_{1})\geq 1$ and $e[b_{2},S_{1}]\geq 1$ both will not lead
to an increase in weight $w(B)$.

Case 4.2. $d_{\mathcal{B}}(b_{1})=d_{\mathcal{B}}(b_{2})=2$.

It can be confirmed that any shared vertex in $B$ has degree at most $3$. Thus
$d(b_{1}),d(b_{2})\leq 3$. We may assume that there does not exist a $6$-$5$
edge in $B$. Then $d(a_{1}),d(a_{2}),d(a_{3})\leq 4$. The number of shared
vertices in $S_{1}$ is $s-2$. It follows that
$w(B)=e(H_{1})+\frac{1}{2}e[H_{1},G\backslash
H_{1}]+\frac{s}{2}\leq\frac{1}{2}[6+4+4\cdot(5-s)+3\cdot
s]+\frac{s}{2}=\frac{30}{2}=\frac{5}{2}\cdot 7-\frac{5}{2}$.

Case 4.3. $d_{\mathcal{B}}(b_{1})=2,d_{\mathcal{B}}(b_{2})=1$.

Similarly, we have $d(b_{1})\leq 3$ and the number of shared vertices in
$S_{1}$ is $s-1$. So $w(B)=e(H_{1})+\frac{1}{2}e[H_{1},G\backslash
H_{1}]+\frac{s}{2}\leq\frac{1}{2}[6+4\cdot 2+4\cdot(4-s)+3\cdot
s]+\frac{s}{2}=\frac{30}{2}=\frac{5}{2}\cdot 7-\frac{5}{2}$.

Case 4.4. $d_{\mathcal{B}}(b_{1})=d_{\mathcal{B}}(b_{2})=1$.

Similarly, we have $w(B)\leq\frac{1}{2}[6+4\cdot 3+4\cdot(3-s)+3\cdot
s]+\frac{s}{2}=\frac{5}{2}\cdot 7-\frac{5}{2}$.

Note that if $s=0$, the equality holds when $B$ is a connected component. But
there must exist a vertex of degree at least $5$ in $B$, which means $B$ is a
block star of $6$-$6$ edge or $6$-$5$ edge.

Case 5. $B$ is a $5$-$5$ edge.

This case is crucial for the proof and is also the most complex part. Let $uv$
be the $5$-$5$ edge. There exist at least $3$ triangles sitting on the edge
$uv$, otherwise an $S_{3,3}$ is found. We distinguish the cases based on the
number of triangles sitting on $uv$.

Case 5.1. There are $4$ triangles sitting on $uv$.

Let $a_{1},a_{2},a_{3}$ and $a_{4}$ be the vertices adjacent to both $u$ and
$v$, as shown in Figure 5$(a)$. Let $S=\\{u,v,a_{1},a_{2},a_{3},a_{4}\\}$,
$S_{1}=\\{a_{1},a_{2},a_{3},a_{4}\\}$ and $H_{1}=G[S_{1}]$.

Figure 5: The star-block $B$ is a $5$-$5$ edge on which four triangles
sitting.

Note that each vertex in $S_{1}$ can have at most one neighbor outside of $B$,
thus $s^{\prime}=0$ and $\mathbf{1}_{B}=0$.

Assume that $e(H_{1})=0$, then $w(B)\leq 9+\frac{4}{2}+\frac{s}{2}$. If $s=0$,
we have $w(B)\leq 11<\frac{5}{2}\cdot 6-\frac{5}{2}$. If $s=1,2$, then $t\geq
2$. It follows $w(B)\leq 12<\frac{5}{2}\cdot 6-\frac{5}{t}$. If $s=3,4$, then
$t\geq 3$, which implies $w(B)\leq 13<\frac{5}{2}\cdot 6-\frac{5}{t}$.

Case 5.1.1. $e(H_{1})=1$.

Without loss of generality, we may assume that $a_{1}a_{4}\in E(G)$. Then
$a_{1},a_{4}$ can not be shared vertices and $s\leq 2$. If there exists one
vertex of $S_{1}$ has no neighbor outside, then $w(B)\leq
10+\frac{3}{2}+\frac{s}{2}\leq\frac{5}{2}\cdot 6-\frac{5}{2}.$ Thus each
vertex of $S_{1}$ has exactly one neighbor outside. Let
$a_{1}a^{\prime}_{1}\in E(G)$ and $B^{\prime}=B+a^{\prime}_{1}$. Then
$\mathcal{B}^{\prime}$ is the corresponding refinement. It is easy to see that
$d(a^{\prime}_{1})\leq 4$, otherwise there exists an $S_{3,3}$.

When $d(a^{\prime}_{1})=4$, we have $a^{\prime}_{1}a_{4}\in E(G)$, as shown in
Figure 5$(b)$, otherwise an $S_{3,3}$ is found. If
$d_{\mathcal{B}^{\prime}}(a^{\prime}_{1})=2$, we have $t\geq 2$ when $s=0$ and
$t\geq 3$ when $s=1,2$. ($s$ is the number of shared vertices in $B$). It
follows that $w(B^{\prime})\leq
12+\frac{4}{2}+\frac{s}{2}+\frac{1}{4}<\frac{5}{2}\cdot 7-\frac{5}{t}$. If
$d_{\mathcal{B}^{\prime}}(a^{\prime}_{1})=1$, then $w(B^{\prime})\leq
12+\frac{4}{2}+\frac{s}{2}\leq\frac{5}{2}\cdot 7-\frac{5}{2}$.

When $d(a^{\prime}_{1})\leq 3$, the star-block is shown in Figure 5$(c)$. Then
$w(B^{\prime})\leq 11+\frac{5}{2}+\frac{s+1}{2}+\mathbf{1}_{B^{\prime}}$. If
$d_{\mathcal{B}^{\prime}}(a^{\prime}_{1})=3$, then $w(B^{\prime})\leq
11+\frac{5}{2}+\frac{s+1}{2}+1<\frac{5}{2}\cdot 7-1$. If
$d_{\mathcal{B}^{\prime}}(a^{\prime}_{1})\leq 2$ then $w(B^{\prime})\leq
11+\frac{5}{2}+\frac{s+1}{2}\leq\frac{5}{2}\cdot 7-\frac{5}{2}$. Specially, if
there is no shared vertex in $G$, we have
$w(B^{\prime})\leq\frac{27}{2}=\frac{5}{2}\cdot 7-4$.

Case 5.1.2. $e(H_{1})=2$.

There are two distinct non-isomorphic subgraphs, as shown in Figure 6$(a,b)$.

Figure 6: The star-block $B$ is a $5$-$5$ edge with $e(H_{1})=2$.

(I). $a_{1}a_{2},a_{2}a_{3}\in E(G)$. It is easy to get $s\leq 1$.

If $d(a_{2})=5$, then there exists an edge $a_{2}a^{\prime}_{2}\in E(G)$ such
that $d(a^{\prime}_{2})\leq 3$. Let $B^{\prime}=B+a^{\prime}_{2}$ and
$\mathcal{B}^{\prime}$ be the corresponding refinement. If
$d_{\mathcal{B}^{\prime}}(a^{\prime}_{2})=3$, then $w(B^{\prime})\leq
12+\frac{5}{2}+\frac{s+1}{2}+1\leq\frac{5}{2}\cdot 7-1$. If
$d_{\mathcal{B}^{\prime}}(a^{\prime}_{2})=2$, then $t\geq 2$ when $s=0$ and
$t\geq 3$ when $s=1$. It follows $w(B^{\prime})\leq
12+\frac{5}{2}+\frac{s+1}{2}\leq\frac{5}{2}\cdot 7-\frac{5}{t}$. If
$d_{\mathcal{B}^{\prime}}(a^{\prime}_{2})=1$, we have $w(B^{\prime})\leq
12+\frac{5}{2}+\frac{s}{2}\leq\frac{5}{2}\cdot 7-\frac{5}{2}$. Specially, if
$s=0$, we have $w(B^{\prime})\leq 12+\frac{5}{2}=\frac{5}{2}\cdot 7-3$.

Next we may assume that $d(a_{2})\leq 4$. If there exists one vertex in
$\\{a_{1},a_{3},a_{4}\\}$ having no neighbor outside, then $w(B)\leq
11+\frac{2}{2}+\frac{s}{2}\leq\frac{5}{2}\cdot 6-\frac{5}{2}$. Specially, if
$s=0$, we have $w(B)\leq 11+\frac{2}{2}=\frac{5}{2}\cdot 6-3$. So we may
assume that each vertex of $\\{a_{1},a_{3},a_{4}\\}$ has a neighbor in
$G\backslash B$. Let $a_{3}a^{\prime}_{3}\in E(G)$ and
$B^{\prime}=B+a^{\prime}_{3}$ with $d(a^{\prime}_{3})\leq 3$. We have
$w(B^{\prime})$ satisfies the upper bound. The proof is similar to
$a^{\prime}_{2}$ mentioned above, so we do not elaborate further here.

(II). $a_{1}a_{4},a_{2}a_{3}\in E(G)$. It is confirmed that $s=0$.

If each vertex in $S_{1}$ has no neighbor outside of $B$, then
$w(B)=11=\frac{5}{2}\cdot 6-4$. Let $a_{2}a^{\prime}_{2}\in E(G)$ and
$B^{\prime}=B+a^{\prime}_{2}$. Note that $d(a^{\prime}_{2})\leq 4$.

When $d(a^{\prime}_{2})=4$, we have $|N(a^{\prime}_{2})\cap S_{1}|=2$. Then
$a^{\prime}_{2}$ must be adjacent to $a_{3}$, otherwise an $S_{3,3}$ is found.
If $a_{4}$ has no neighbor outside, then $w(B^{\prime})\leq
13+\frac{3}{2}+\frac{1}{4}<\frac{5}{2}\cdot 7-\frac{5}{2}$. Let
$a_{4}a^{\prime}_{4}\in E(G)$, $B^{*}=B^{\prime}+a^{\prime}_{4}$ and
$\mathcal{B}^{*}$ is the corresponding refinement of $\mathcal{B}^{\prime}$.
Similarly, when $d(a^{\prime}_{4})=4$, we have $|N(a^{\prime}_{2})\cap
S_{1}|=2$, as shown in Figure 6$(c)$. If
$d_{\mathcal{B}^{*}}(a^{\prime}_{2})=d_{\mathcal{B}^{*}}(a^{\prime}_{4})=2$,
then $t\geq 3$. It follows $w(B^{*})\leq
15+\frac{4}{2}+\frac{2}{4}<\frac{5}{2}\cdot 8-\frac{5}{t}$. Otherwise,
$w(B^{*})\leq 15+\frac{4}{2}+\frac{1}{4}<\frac{5}{2}\cdot 8-\frac{5}{2}$. When
$d(a^{\prime}_{4})=3$, it can be proved similarly. If
$d_{\mathcal{B}^{*}}(a^{\prime}_{4})=3$, we get $w(B^{*})\leq
14+\frac{5}{2}+\frac{1}{2}+\frac{1}{4}+1<\frac{5}{2}\cdot 8-1$. If
$d_{\mathcal{B}^{*}}(a^{\prime}_{4})=2$, then $w(B^{*})\leq
14+\frac{5}{2}+\frac{1}{2}+\frac{1}{4}<\frac{5}{2}\cdot 8-\frac{5}{2}$. If
$d_{\mathcal{B}^{*}}(a^{\prime}_{4})=1$, we have $w(B^{*})\leq
14+\frac{5}{2}+\frac{1}{4}<\frac{5}{2}\cdot 8-3$. Specially, if there is no
shared vertex in $G$, we have $w(B^{*})\leq 14+\frac{5}{2}=\frac{5}{2}\cdot
8-\frac{7}{2}$.

When $d(a^{\prime}_{2})\leq 3$, we can also analyze it based on
$d_{\mathcal{B}^{\prime}}(a^{\prime}_{2})$. If
$d_{\mathcal{B}^{\prime}}(a^{\prime}_{2})=3$, we have $w(B^{\prime})\leq
12+\frac{5}{2}+\frac{1}{2}+1<\frac{5}{2}\cdot 7-1$. If
$d_{\mathcal{B}^{\prime}}(a^{\prime}_{2})=2$, then $w(B^{\prime})\leq
12+\frac{5}{2}+\frac{1}{2}=\frac{5}{2}\cdot 7-\frac{5}{2}$. If
$d_{\mathcal{B}^{\prime}}(a^{\prime}_{2})=1$, it follows $w(B^{\prime})\leq
12+\frac{5}{2}=\frac{5}{2}\cdot 7-3$.

Case 5.1.3. $e(H_{1})=3$.

Let $a_{1}a_{2},a_{2}a_{3},a_{1}a_{4}\in E(G)$, as shown in Figure 7.

Figure 7: The star-block $B$ is a $5$-$5$ edge with $e(H_{1})=3$.

Assume that $d(a_{4})=4$. Let $a_{4}a^{\prime}_{4}\in E(G)$ and
$B^{\prime}=B+a^{\prime}_{4}$. Note that $d(a^{\prime}_{4})=3$, otherwise we
find an $S_{3,3}$. If $d(a_{2})=4$ and $d(a_{3})=3$, only $a_{1}$ can have a
neighbor outside of $B^{\prime}$. When
$d_{\mathcal{B}^{\prime}}(a^{\prime}_{4})=3$, then $w(B^{\prime})\leq
13+\frac{3}{2}+\frac{1}{2}+1<\frac{5}{2}\cdot 7-1$. When
$d_{\mathcal{B}^{\prime}}(a^{\prime}_{4})=2$, then $w(B^{\prime})\leq
13+\frac{3}{2}+\frac{1}{2}=\frac{5}{2}\cdot 7-\frac{5}{2}$. When
$d_{\mathcal{B}^{\prime}}(a^{\prime}_{4})=1$, it follows $w(B^{\prime})\leq
13+\frac{3}{2}=\frac{5}{2}\cdot 7-3$. If $d(a_{2})=5$ and $d(a_{3})=3$, there
exists an edge $a_{2}a^{\prime}_{2}\in E(G)$ with $d(a^{\prime}_{2})=3$. Let
$B^{*}=B^{\prime}+a^{\prime}_{2}$ and $\mathcal{B}^{*}$ is the corresponding
refinement of $\mathcal{B}^{\prime}$. When
$d_{\mathcal{B}^{*}}(a^{\prime}_{4})=3$ or
$d_{\mathcal{B}^{*}}(a^{\prime}_{2})=3$, we get $w(B^{*})\leq
14+\frac{5}{2}+\frac{2}{2}+1<\frac{5}{2}\cdot 8-1$. When
$d_{\mathcal{B}^{*}}(a^{\prime}_{4}),d_{\mathcal{B}^{*}}(a^{\prime}_{2})\leq
2$, then $w(B^{*})\leq 14+\frac{5}{2}+\frac{2}{2}=\frac{5}{2}\cdot
8-\frac{5}{2}$. when
$d_{\mathcal{B}^{*}}(a^{\prime}_{4})=d_{\mathcal{B}^{*}}(a^{\prime}_{2})=1$,
we have $w(B^{*})\leq 14+\frac{5}{2}=\frac{5}{2}\cdot 8-\frac{7}{2}$. If
$d(a_{3})=4$, we have an edge $a_{3}a^{\prime}_{3}$ with
$d(a^{\prime}_{3})=3$. Let $B^{*}=B^{\prime}+a^{\prime}_{3}$. The subsequent
discussion is analogous, so it will not be reiterated here. We can prove that
$B^{*}$ or some other star-block refined from $B^{*}$ satisfies the upper
bound.

Hence we have $d(a_{3})=d(a_{4})=3$ by the symmetry of these two vertices.

Next we show that $d(a_{1})\leq 4$ or $d(a_{2})\leq 4$. If
$d(a_{1})=d(a_{2})=5$, there exist edges
$a_{1}a^{\prime}_{1},a_{2}a^{\prime}_{2}$ in $G$ and
$d(a^{\prime}_{1}),d(a^{\prime}_{2})\leq 3$. If $a^{\prime}_{1}\neq
a^{\prime}_{2}$, let $B^{\prime}=B+a^{\prime}_{1}+a^{\prime}_{2}$. Then
$w(B^{\prime})\leq
14+\frac{4}{2}+\frac{2}{2}+\mathbf{1}_{B^{\prime}}=\frac{5}{2}\cdot
8-3+\mathbf{1}_{B^{\prime}}$, satisfying the upper bound. If
$a^{\prime}_{1}=a^{\prime}_{2}$, then let $B^{\prime}=B+a^{\prime}_{1}$. We
have $d_{\mathcal{B^{\prime}}}(a^{\prime}_{1})\leq 2$ and $w(B^{\prime})\leq
14+\frac{1}{2}+\frac{1}{2}=\frac{5}{2}\cdot 7-\frac{5}{2}$. Specially, if
there is no shared vertex in $G$, we have $w(B^{\prime})\leq
14+\frac{1}{2}=\frac{5}{2}\cdot 7-3$.

Now we have $d(a_{1})\leq 4$ or $d(a_{2})\leq 4$, thus $w(B)\leq
12+\frac{1}{2}=\frac{5}{2}\cdot 6-\frac{5}{2}$.

Case 5.2. There are $3$ triangles sitting on $uv$.

Let $a_{1}$, $a_{2}$ and $a_{3}$ be the vertices adjacent to both $u$ and $v$.
Let $b_{1}$ be the vertex only adjacent to $u$ and $b_{2}$ be the vertex only
adjacent to $v$, see Figure 8$(a)$ as an example. Let
$S=\\{u,v,a_{1},a_{2},a_{3},b_{1},b_{2}\\}$, $S_{1}=\\{a_{1},a_{2},a_{3}\\}$,
$S_{2}=\\{b_{1},b_{2}\\}$ and $H_{i}=G[S_{i}]$ for $i\in\\{1,2\\}$.

Figure 8: The star-block $B$ is a $5$-$5$ edge on which three triangles
sitting.

Recall that each vertex in $S_{1}$ can have at most one neighbor outside of
$B$. And it can be checked that $d(b_{1}),d(b_{2})\leq 4$. If
$d(b_{1})=d(b_{2})=4$, then $b_{1},b_{2}$ both have two neighbors in $S_{1}$.

We may assume that there does not exist vertex of degree $6$ in $B$ by the
cases discussed above. Let $s_{0}$ denote the number of shared vertices in
$S_{1}$.

Case 5.2.1. $d(b_{1}),d(b_{2})\leq 3$.

(I). $d_{\mathcal{B}}(b_{1})=3,d_{\mathcal{B}}(b_{2})\leq 3$.

We have $d(b_{1})=3$ and $v(B)=7$. Note that $b_{1}$ has no neighbor in
$S_{1}$. If $b_{2}a_{1}\in E(G)$, then $a_{1}$ can not have a neighbor outside
of $B$. So it is determined that $d(a_{1})\leq 5$ and $d(a_{2}),d(a_{3})\leq
4$. Moreover any shared vertex in $S_{1}$ has degree $3$.

It is obtained $w(B)=\frac{1}{2}\sum\limits_{v\in
B}d(v)+\frac{s}{2}+\mathbf{1}_{B}\leq\frac{1}{2}(5\cdot 3+4\cdot
2-s_{0}+3\cdot 2)+\frac{2+s_{0}}{2}+1=\frac{5}{2}\cdot 7-1$.

(II). $d_{\mathcal{B}}(b_{1})=d_{\mathcal{B}}(b_{2})=2$.

Assume that $b_{1}b_{2}\in E(G)$. Since $b_{1},b_{2}$ are shared vertices, it
is easy to know that $b_{1},b_{2}$ is not adjacent to any vertex in $S_{1}$
and $d(b_{1})=d(b_{2})=3$.

There are two possible planar embeddings.

For the first planar embedding $(\alpha)$, as shown in Figure 8$(b)$, we have
$a_{1}a_{3}\notin E(G)$, $d(a_{1}),d(a_{3})\leq 4$ and $d(a_{2})\leq 3$. When
$s_{0}\leq 1$, $w(B)\leq\frac{1}{2}(5\cdot 2+4\cdot 2+3\cdot
3)+\frac{s_{0}+2}{2}\leq\frac{30}{2}=\frac{5}{2}\cdot 7-\frac{5}{2}$. When
$s_{0}\geq 2$, $w(B)\leq\frac{1}{2}(5\cdot 2+4\cdot 2+3\cdot
3-s_{0}+1)+\frac{s_{0}+2}{2}=\frac{5}{2}\cdot 7-\frac{5}{2}$.

For the second planar embedding $(\beta)$, as shown in Figure 8$(c)$, there
must exist a vertex in $S_{1}$, say $a_{2}$, such that the vertex $a_{2}$ is
in a different region from vertices $b_{1}$ and $b_{2}$. If $d(a_{1})=5$,
there exists an edge $a_{1}a^{\prime}_{1}\in E(G)$ such that
$d(a^{\prime}_{1})\leq 3$, otherwise an $S_{3,3}$ is found. Let
$B^{\prime}=B+a^{\prime}_{1}$ and $\mathcal{B}^{\prime}$ is the corresponding
refinement of $\mathcal{B}$. It is obtained that
$w(B^{\prime})\leq\frac{1}{2}(5\cdot 3+4\cdot 2-s_{0}+3\cdot
3)+\frac{3+s_{0}}{2}+\mathbf{1}_{B^{\prime}}=\frac{5}{2}\cdot
8-\frac{5}{2}+\mathbf{1}_{B^{\prime}}$, which satisfies the upper bound.
Specially, if $d_{\mathcal{B}^{\prime}}(a^{\prime}_{1})=1$, we have
$w(B^{\prime})\leq 17=\frac{5}{2}\cdot 8-3$.

Therefore we can conclude that $b_{1}b_{2}\notin E(G)$. Next we will show that
$d(a_{2}),d(a_{3})\leq 4$. In fact, if there exists a vertex of degree $5$,
say $a_{3}$, then $a_{3}b_{1},a_{3}b_{2},a_{1}a_{3}\in E(G)$, which implies
$d(a_{2})\leq 4$ and $a_{1},a_{3}$ are not shared vertices. If $d(a_{1})=5$,
then $a_{1}a_{2}\in E(G)$, as shown in Figure 9$(a)$. There is an $S_{3,3}$.
So we get $d(a_{1})\leq 4$. If $d(a_{2})=4$, there is an edge
$a_{2}a^{\prime}_{2}$ with $d(a^{\prime}_{2})\leq 3$. Let
$B^{\prime}=B+a^{\prime}_{2}$ and $\mathcal{B}^{\prime}$ is the corresponding
refinement of $\mathcal{B}$. We have $w(B^{\prime})\leq\frac{1}{2}(5\cdot
3+4\cdot 2+3\cdot 3)+\frac{2}{2}+\mathbf{1}_{B^{\prime}}=\frac{5}{2}\cdot
8-3+\mathbf{1}_{B^{\prime}}$, satisfying the bound. Thus $d(a_{1})\leq 4$ and
$d(a_{2})\leq 3$. It follows $w(B)\leq\frac{1}{2}(5\cdot 3+4\cdot
1-s_{0}+3\cdot 3)+\frac{2+s_{0}}{2}=\frac{5}{2}\cdot 7-\frac{5}{2}$.

Figure 9: The star-block $B$ is a $5$-$5$ edge with $b_{1}b_{2}\notin E(G)$.

Now we may assume that $d(a_{2}),d(a_{3})\leq 4$ by the symmetry of these two
vertices. If $b_{1},b_{2}$ are in different regions, we obtain $t\geq 3$ and
$w(B)\leq\frac{1}{2}(5\cdot 3+4\cdot 2-s_{0}+3\cdot
2)+\frac{s_{0}+2}{2}=\frac{31}{2}<\frac{5}{2}\cdot 7-\frac{5}{t}$.

We assume that $b_{1},b_{2}$ are in the same region. Then there exists a
vertex, say $a_{2}$, located in a different region. If $d(a_{2})=4$, then let
$a_{2}a^{\prime}_{2}\in E(G)$ and $B^{\prime}=B+a^{\prime}_{2}$. It follows
that $w(B^{\prime})\leq\frac{1}{2}(5\cdot 3+4\cdot 2-s_{0}+3\cdot
3)+\frac{s_{0}+3}{2}+\mathbf{1}_{B^{\prime}}=\frac{5}{2}\cdot
8-\frac{5}{2}+\mathbf{1}_{B^{\prime}}$, which satisfies the upper bound.
Specially, if $d_{\mathcal{B}^{\prime}}(a^{\prime}_{2})=1$,
$w(B^{\prime})\leq\frac{5}{2}\cdot 8-3$.

Hence we have $d(a_{2})\leq 3$. If $d(a_{1})=5$, there are only two possible
planar embeddings, shown in Figure 9$(b,c)$. For the first planar embedding
$(\alpha)$, it is known that $d(a_{3})\leq 3$ and $s_{0}\leq 1$. It follows
$w(B)\leq\frac{1}{2}(5\cdot 3+3\cdot 4)+\frac{s_{0}+2}{2}\leq\frac{5}{2}\cdot
7-\frac{5}{2}$. For the second planar embedding $(\beta)$, we know $s_{0}=0$.
Then $w(B)\leq\frac{1}{2}(5\cdot 3+4+3\cdot 3)+\frac{2}{2}\leq\frac{5}{2}\cdot
7-\frac{5}{2}$.

Now we have $d(a_{1}),d(a_{3})\leq 4$ and $d(a_{2})\leq 3$. Hence
$w(B)\leq\frac{1}{2}[5\cdot 2+4\cdot 2+3\cdot
3-\mathbf{1}_{s_{0}}\cdot(s_{0}-1)]+\frac{s_{0}+2}{2}\leq\frac{30}{2}=\frac{5}{2}\cdot
7-\frac{5}{2}$, where $\mathbf{1}_{s_{0}}$ is the characteristic function.

(III). $d_{\mathcal{B}}(b_{1})=2,d_{\mathcal{B}}(b_{2})=1$.

We demonstrate that the proof here is essentially the same as the
aforementioned discussion, and there is a relationship between the weights of
them.

Note that the calculations for $w(B)$ above were based on the assumption that
all vertices in $B$ have degree at least $3$. As shown in Figure 8$(a)$, if
$d(b_{2})\leq 2$, then the star-block structure here is a subgraph of a
certain subcase, say $B^{\prime}$, in (II). So it is a process that ensures a
decrease in weight. We have $w(B)\leq w(B^{\prime})-\frac{1}{2}$. By verifying
the above results sequentially, we obtain $w(B)$ satisfies the upper bound.

Hence we can assume that $d(b_{2})=3$. If $b_{2}$ has a neighbor outside of
$B$, it is checked that the subgraph structure is the same as previously
discussed in (II). Note that $b_{2}$ is not a shared vertex anymore. Comparing
to the previous value, the $w(B)$ here will be reduced by $1/2$. This means
$w(B)$ satisfies the upper bound.

Thus $N(b_{2})\subset V(B)$. Note that $b_{1}$ has at least one neighbor
outside by $d_{\mathcal{B}}(b_{1})=2$. There must exist a vertex in $S_{1}$,
say $a_{1}$, which is adjacent to $b_{2}$ and not adjacent to $b_{1}$. Note
that $a_{1}$ can not have a neighbor outside of $B$.

Now we make some modifications to this star-block. Let us cut the edge
$a_{1}b_{2}$. And suppose that $a_{1}$ has a neighbor outside of $B$ and
$b_{2}$ has a neighbor outside too. The modified star-block is denoted as
$B^{\prime}$. If $d_{\mathcal{B}^{\prime}}(b_{2})=2$, $B^{\prime}$ is a
subcase in (II) and $w_{0}(B)=w_{0}(B^{\prime})$. The difference between $B$
and $B^{\prime}$ is the number of shared vertices they contain. It can be
checked that $w(B)\leq w(B^{\prime})-\frac{1}{2}$.

(IV). $d_{\mathcal{B}}(b_{1})=d_{\mathcal{B}}(b_{2})=1$.

As in the previous discussion, we can categorize the situation here as one of
the aforementioned subcases in (III).

If $b_{1}$ has a neighbor outside, it is easy to see that the star-block here
corresponds to a certain subgraph in (III).

Assume that $a_{1}b_{1}$ is an edge. We make some modifications to $B$ by
cutting the edge and keeping the other connections of $a_{1}$. Now suppose
$b_{1}$ have a neighbor outside and $d_{\mathcal{B}^{\prime}}(b_{1})=2$, and
let $B^{\prime}$ denote the new star-block. It is known that $B^{\prime}$ is a
star-block in (III) and $w(B)=w(B^{\prime})$.

Case 5.2.2. $d(b_{1})=4,d(b_{2})\leq 3$.

Since $d(b_{1})=4$, $b_{1}$ has two neighbors, say $a_{1},a_{3}$, in $S_{1}$,
as shown in Figure 10$(a)$. Then $a_{1},a_{3},b_{1}$ are not shared vertices
and $a_{2},b_{1}$ are in different regions. This means that $d(a_{2})\leq 4$
and $a_{1},a_{3}$ have no neighbor outside of $B$. Let $s_{0}$ be the number
of shared vertices in $S_{1}$. Then $s_{0}\leq 1$.

Figure 10: A $5$-$5$ edge with $d(b_{1})=4,d(b_{2})\leq 3$.

(I). $d_{\mathcal{B}}(b_{2})=3$.

Note that $b_{2}$ is not adjacent to any vertex in $S_{1}\cup\\{b_{1}\\}$. We
have $d(a_{3})\leq 4$. Thus $w(B)\leq\frac{1}{2}(5\cdot 3+4\cdot
3-s_{0}+3\cdot 1)+\frac{s_{0}+1}{2}+\mathbf{1}_{B}=\frac{5}{2}\cdot 7-1$.

(II). $d_{\mathcal{B}}(b_{2})=2$.

If $a_{1}a_{3}\in E(G)$, $b_{1},b_{2}$ are in different regions. Then there
exists an edge $b_{1}b^{\prime}_{1}$ such that $d(b^{\prime}_{1})\leq 3$. Let
$B^{\prime}=B+b^{\prime}_{1}$.

When $d_{\mathcal{B}^{\prime}}(b^{\prime}_{1})=3$, we have
$w(B^{\prime})\leq\frac{1}{2}(5\cdot 4+4\cdot 2-s_{0}+3\cdot
2)+\frac{s_{0}+2}{2}+\mathbf{1}_{B^{\prime}}\leq\frac{38}{2}=\frac{5}{2}\cdot
8-1$.

When $d_{\mathcal{B}^{\prime}}(b^{\prime}_{1})=2$, we obtain $t\geq 3$. Then
$w(B^{\prime})\leq\frac{36}{2}\leq\frac{5}{2}\cdot 8-\frac{5}{t}$.

When $d_{\mathcal{B}^{\prime}}(b^{\prime}_{1})=1$, it follows
$w(B^{\prime})\leq\frac{1}{2}(5\cdot 4+4\cdot 2-s_{0}+3\cdot
2)+\frac{s_{0}+1}{2}\leq\frac{35}{2}=\frac{5}{2}\cdot 8-\frac{5}{2}$.

Now we may assume that $a_{1}a_{3}\notin E(G)$. It is easy to see that
$d(a_{3})\leq 4$. Recall that $b_{2}$ has a neighbor outside of $B$. This
implies that $d(a_{1})+d(a_{3})\leq 8$. So $w(B)\leq\frac{1}{2}(5\cdot
2+8+4\cdot 2-s_{0}+3\cdot 1)+\frac{s_{0}+1}{2}=\frac{5}{2}\cdot
7-\frac{5}{2}$.

(III). $d_{\mathcal{B}}(b_{2})=1$.

We have $w(B)\leq\frac{1}{2}(5\cdot 4+4\cdot
2-s_{0}+3)+\frac{s_{0}}{2}=\frac{5}{2}\cdot 7-2$. The equality holds when
$d(b_{2})=3$, $d(a_{1})=d(a_{3})=5$ and $d(a_{2})=4$, otherwise we are done.
It can be checked that $a_{1}$ is adjacent to $a_{3}$ as shown in 10$(b)$,
which implies $b_{1}$ and $b_{2}$ are in different regions. So there exists an
edge $b_{1}b^{\prime}_{1}$ such that $d(b^{\prime}_{1})\leq 3$. Let
$B^{\prime}=B+b^{\prime}_{1}$. Then $w(B^{\prime})\leq\frac{1}{2}(5\cdot
4+4\cdot 2-s_{0}+3\cdot
2)+\frac{s_{0}+1}{2}+\mathbf{1}_{B^{\prime}}=\frac{5}{2}\cdot
8-\frac{5}{2}+\mathbf{1}_{B^{\prime}}$, which satisfies the upper bound.

Case 5.2.3. $d(b_{1})=d(b_{2})=4$.

It is easy to see that $d_{\mathcal{B}}(b_{1})=d_{\mathcal{B}}(b_{2})=1$ and
$s_{0}\leq 1$. Note that $b_{1},b_{2}$ both have two neighbors in $S_{1}$.
Without loss of generality, we assume that $a_{1},a_{3}$ are neighbors of
$b_{1}$. Thus $d(a_{2})\leq 4$ and $a_{1},a_{3}$ have no neighbor outside of
$B$.

(I). $N(b_{1})=N(b_{2})$.

The graph of this situation is shown in Figure 11$(a)$. If $b_{1}b_{2}\in
E(G)$, then $a_{1}a_{3}\notin E(G)$, which implies $d(a_{3})\leq 4$. If
$d(a_{2})\leq 3$, we have $w(B)\leq\frac{1}{2}(5\cdot 3+4\cdot 3-s_{0}+3\cdot
1)+\frac{s_{0}}{2}=\frac{5}{2}\cdot 7-\frac{5}{2}$. If $d(a_{2})=4$, there
exist edges $a_{1}a_{2},a_{2}a^{\prime}_{2}\in E(G)$ with
$d(a^{\prime}_{2})\leq 3$. Note that $s_{0}=0$. Let
$B^{\prime}=B+a^{\prime}_{2}$ and $\mathcal{B^{\prime}}$ be the corresponding
refinement of $\mathcal{B}$. Hence $w(B^{\prime})\leq\frac{1}{2}(5\cdot
3+4\cdot 4+3\cdot 1)+\frac{1}{2}+\mathbf{1}_{B^{\prime}}=\frac{5}{2}\cdot
8-\frac{5}{2}+\mathbf{1}_{B^{\prime}}$, which satisfies the upper bound.

Now we assume that $b_{1}b_{2}\notin E(G)$. There exist edges
$b_{1}b^{\prime}_{1},b_{2}b^{\prime}_{2}$ with
$d(b^{\prime}_{1}),d(b^{\prime}_{2})\leq 3$. If
$b^{\prime}_{1}=b^{\prime}_{2}$, then $a_{1}a_{3}\notin E(G)$, which implies
$d(a_{3})\leq 4$. Let $B^{\prime}=B+b^{\prime}_{1}$. Note that
$d_{\mathcal{B}^{\prime}}(b^{\prime}_{1})\leq 2$. Thus
$w(B^{\prime})\leq\frac{1}{2}(5\cdot 3+4\cdot 4-s_{0}+3\cdot
1)+\frac{s_{0}+1}{2}=\frac{5}{2}\cdot 8-\frac{5}{2}$. If $b^{\prime}_{1}\neq
b^{\prime}_{2}$, let $B^{\prime}=B+b^{\prime}_{1}+b^{\prime}_{2}$. Now we show
that $d(a_{1})+d(a_{2})+d(a_{3})\leq 13$. In fact, if $a_{1}a_{3}\in E(G)$,
then $d(a_{1}),d(a_{3})\geq 5$. Since we assume that there does not exist a
star-block like $6$-$5$ edge, we get $d(a_{1})=d(a_{3})=5$. Then
$a_{1}a_{2}\notin E(G)$, which implies $d(a_{2})\leq 3$. If $a_{1}a_{3}\notin
E(G)$, it is easy to know $d(a_{1})\leq 5$, $d(a_{2})\leq 4$, $d(a_{1})\leq
4$. Thus it is obtained $d(a_{1})+d(a_{2})+d(a_{3})\leq 13$. Therefore we have
$w(B^{\prime})\leq\frac{1}{2}(5\cdot 2+13+4\cdot 2+3\cdot
2)+\frac{s_{0}+2}{2}+\mathbf{1}_{B^{\prime}}\leq\frac{5}{2}\cdot
9-\frac{5}{2}+\mathbf{1}_{B^{\prime}}$, which satisfies the upper bound.

Figure 11: A $5$-$5$ edge with $d(b_{1})=d(b_{2})=4$.

(II). $N(b_{1})\neq N(b_{2})$.

Assume that $\\{a_{1},a_{2}\\}\subseteq N(b_{2})$, as shown in Figure 11$(b)$.
Obviously, $b_{1},b_{2}$ are in different regions and any vertex in $S_{1}$
has no neighbor outside of $B$. That means $s_{0}=0$. By the assumption that
there is no star-block like $6$-$5$ edge, so $a_{1}a_{2},a_{1}a_{3}$ can not
both be edges in $G$. It is obtained $d(a_{1})+d(a_{2})+d(a_{3})\leq 12$. Let
$b_{1}b^{\prime}_{1},b_{2}b^{\prime}_{2}\in E(G)$ and
$B^{\prime}=B+b^{\prime}_{1}+b^{\prime}_{2}$, we have
$w(B^{\prime})\leq\frac{1}{2}(5\cdot 2+12+4\cdot 2+3\cdot
2)+\frac{2}{2}+\mathbf{1}_{B^{\prime}}\leq\frac{5}{2}\cdot
9-\frac{5}{2}+\mathbf{1}_{B^{\prime}}$, which satisfies the upper bound.

Case 6. $B$ is a $5$-$4$-$5$ path.

We may assume that there are no star-blocks like $6$-$5$ edge, $5$-$5$ edge.
This means that all $5$-degree vertices form an independent set. Let $u,v,w$
be the vertices in the $5$-$4$-$5$ path and $d(v)=4$. Consider the $5$-$4$
edge $uv$. Since $G$ is $S_{3,3}$-free, the number of triangles sitting on
$uv$ is at least $2$. There are three possible subgraphs, seen in Figure 12.

Figure 12: All $5$-$4$-$5$ paths.

Now we consider the subgraphs in turn.

For the subgraph $(a)$, let
$S=\\{u,v,w,a_{1},a_{2},b_{1},b_{2},b_{3},b_{4}\\}$, $S_{1}=\\{a_{1},a_{2}\\}$
and $S_{2}=\\{b_{1},b_{2},b_{3},b_{4}\\}$. It can be determined that each
vertex in $S_{2}$ can have at most two neighbors in $S_{2}\cup G\backslash B$
and any vertex in $S_{1}$ has no neighbor in $S_{2}\cup G\backslash B$. Thus
$d(b_{1}),d(b_{2}),d(b_{3}),d(b_{4})\leq 3$. It follows
$w(B)\leq\frac{1}{2}(5\cdot 2+4\cdot 3+3\cdot
4)+\frac{4}{2}+\mathbf{1}_{B}=\frac{5}{2}\cdot 9-\frac{7}{2}+\mathbf{1}_{B}$.
Specially, if there does not exist any shared vertex, we have
$w(B)\leq\frac{5}{2}\cdot 9-\frac{11}{2}$.

Similarly, for the subgraph $(b)$, we have $w(B)\leq\frac{1}{2}(5\cdot
2+4\cdot 4+3\cdot 2)+\frac{3}{2}+\mathbf{1}_{B}=\frac{5}{2}\cdot
8-\frac{5}{2}+\mathbf{1}_{B}$.

For the subgraph $(c)$, $b_{1},b_{2}$ both have two neighbors in $B$. Hence,
$d_{\mathcal{B}}(b_{1}),d_{\mathcal{B}}(b_{2})\leq 2$, which implies
$\mathbf{1}_{B}=0$. If $d_{\mathcal{B}}(b_{1})=2$ or
$d_{\mathcal{B}}(b_{2})=2$, then $w(B)\leq\frac{1}{2}(5\cdot 2+4\cdot 3+3\cdot
2)+\frac{2}{2}=\frac{5}{2}\cdot 7-\frac{5}{2}$. If
$d_{\mathcal{B}}(b_{1})=d_{\mathcal{B}}(b_{2})=1$, we have
$w(B)\leq\frac{1}{2}(5\cdot 2+4\cdot 5)=\frac{5}{2}\cdot 7-\frac{5}{2}$.

In summary, $w(B)$ satisfies the upper bound.

Case 7. $B$ is a $5$-$4^{-}$ star.

Let $u$ be the vertex of degree 5, we claim that any vertex $v\in N(u)$ can
not be a shared vertex. In fact, there are at least $2$ triangles sitting on
the edge $5$-$4$, otherwise an $S_{3,3}$ is found in $G$. This means that $v$
have at least three neighbors in $N[u]\backslash v$. By the definition of
star-block, $v$ can not be a shared vertex.

Therefore we have $w(B)=\frac{1}{2}(5+5\cdot 4)=\frac{5}{2}\cdot
6-\frac{5}{2}$.

##  4 Proof of Lemma 2.2

Since $G$ contains only one star-block $B$, then all vertices outside of the
star-block have degree at most $4$. There does not exist any shared vertex,
which implies $w(B)=w_{0}(B)$.

If $G$ is disconnected, we assume that $G_{1}$ is the component containing the
star-block and $G_{2}=G\backslash G_{1}$. Let $k_{1}=v(G_{1})$,
$k^{\prime}_{1}=v(B)$ and $k_{2}=v(G_{2})$. By Lemma 2.1,
$w(B)\leq\frac{5}{2}v(B)-\frac{5}{2}$. It follows that
$e(G_{1})=w(G_{1})=w(B)+w(G_{1}\backslash
B)\leq\frac{5}{2}k^{\prime}_{1}-\frac{5}{2}+\frac{1}{2}[4(k_{1}-k^{\prime}_{1})]$.
When $1\leq k_{2}\leq 2$, then $e(G_{2})\leq\frac{5}{2}k_{2}-\frac{5}{2}$.
When $3\leq k_{2}\leq 7$, it is easy to check that $e(G_{2})\leq
3k_{2}-6\leq\frac{5}{2}k_{2}-\frac{5}{2}$. When $k_{2}\geq 8$, we also have
$e(G_{2})=\frac{1}{2}\sum\limits_{v\in
V(G_{2})}d(v)\leq\frac{4k_{2}}{2}\leq\frac{5}{2}k_{2}-\frac{5}{2}$.

Hence, we have

$\displaystyle e(G)$ $\displaystyle=e(G_{1})+e(G_{2})$
$\displaystyle\leq\frac{5}{2}k^{\prime}_{1}-\frac{5}{2}+2(k_{1}-k^{\prime}_{1})+\frac{5}{2}k_{2}-\frac{5}{2}$
$\displaystyle=\frac{5}{2}n-5-\frac{1}{2}(k_{1}-k^{\prime}_{1})$
$\displaystyle\leq\frac{5}{2}n-5.$

Now we can assume that $G$ is connected and we will discuss each type of star-
blocks in turn. Let $k=v(B)$.

Case 1. $G$ contains a $5^{+}$-$3^{-}$ star.

Let $B$ be the $5^{+}$-$3^{-}$ star and $k\geq 6$. It follows
$w(B)\leq\frac{1}{2}[(k-1)+3(k-1)]=2(k-1)\leq\frac{5}{2}k-5$. Thus

$\displaystyle e(G)$ $\displaystyle=w(B)+w(G\backslash B)$
$\displaystyle\leq\frac{5}{2}k-5+\sum\limits_{v\in G\backslash B}d(v)$
$\displaystyle\leq\frac{5}{2}k-5+\frac{1}{2}\cdot 4(n-k)$
$\displaystyle\leq\frac{5}{2}n-5.$

Case 2. $G$ contains a $6$-$6$ edge, or $6$-$5$ edge, or $6$-$4$ edge.

Referring to the final discussion in the proof for Cases $2,3,4$ of Lemma 2.1,
we can deduce that $B$ is the elementary star-block on $7$ vertices.
Furthermore, if $B$ is a connected component, then $n=k=7$ and $e(G)=w(B)\leq
15$. If there are other vertices not contained in this star-block, we have
$w(B)\leq\frac{5}{2}\cdot 7-3$. Then

$\displaystyle e(G)$ $\displaystyle=w(B)+w(G\backslash B)$
$\displaystyle\leq\frac{5}{2}\cdot 7-3+\frac{1}{2}[4(n-7)]$
$\displaystyle=2n+\frac{1}{2}.$

Since $e(G)$ is an integer, we have $e(G)\leq 2n$. This means $e(G)\leq 16$
when $n=8$, $e(G)\leq 18$ when $n=9$ and $e(G)\leq 20$ when $n=10$.
Furthermore, when $n\geq 11$, it follows $e(G)\leq
2n+\frac{1}{2}\leq\frac{5}{2}n-5$.

Case 3. $G$ contains a $5$-$5$ edge.

We will show that there must exist a star-block in $G$ whose weight is not
large.

######  Claim 4.1.

For any type of $5$-$5$ edge except one certain subcase, there exists a star-
block $B$ containing it such that
$w(B)\leq\frac{5}{2}k-\frac{k-1}{2}=2k+\frac{1}{2}$.

###### Proof.

If there are $4$ triangles sitting on $uv$, as shown in Figure 5$(a)$, it can
be determined based on the proof for Case 5.1 of Lemma 2.1.

Then it remains to prove the case that there are $3$ triangles sitting on
$uv$, as shown in Figure 8$(a)$. However, there are too many subcases and it
is tedious to check each one individually. Here, we provide a new proof.

Let $uv$ be the $5$-$5$ edge in $G$. Let
$S=\\{u,v,a_{1},a_{2},a_{3},b_{1},b_{2}\\}$, $S_{1}=\\{a_{1},a_{2},a_{3}\\}$
and $H_{1}=G[S_{1}]$. Without loss of generality, we may assume that there
does not exist vertex of degree $6$ in $G$.

(I). $d(b_{1}),d(b_{2})\leq 3$.

Note that $a_{2},a_{3}$ can not both be the vertices of degree $5$. In fact,
if $d(a_{2})=5$, then $a_{2}b_{1},a_{2}b_{2}\in E(G)$, which implies
$d(a_{3})\leq 4$. Thus $d(a_{2})+d(a_{3})\leq 9$.

Assume $d(a_{3})=5$. If $d(a_{2})\leq 3$, then $w(B)\leq\frac{1}{2}(5\cdot
4+3\cdot 3)=\frac{5}{2}\cdot 7-3$. If $d(a_{2})=4$, there exists an edge
$a_{2}a^{\prime}_{2}\in E(G)$. Let $B^{\prime}=B+a^{\prime}_{2}$. Hence
$w(B^{\prime})\leq\frac{1}{2}(5\cdot 4+4+3\cdot 3)=\frac{5}{2}\cdot
8-\frac{7}{2}$.

Hence we may assume that $d(a_{3})\leq 4$ and $d(a_{2})\leq 4$. Then
$w(B)\leq\frac{1}{2}(5\cdot 3+4\cdot 2+3\cdot 2)=\frac{5}{2}\cdot 7-3$.

(II). $d(b_{1})=4,d(b_{2})\leq 3$.

Note that $b_{1}$ has two neighbors in $S_{1}$, as shown in Figure 10$(a)$.
Assume that $a_{1}b_{1},a_{3}b_{1}\in E(G)$. Then $a_{1},a_{3}$ have no
neighbor outside of $B$ and $b_{1},a_{2}$ are in different regions.

Since $d(b_{1})=4$, there exists an edge $b_{1}b^{\prime}_{1}\in E(G)$ with
$d(b^{\prime}_{1})\leq 3$.

If $b^{\prime}_{1}=b_{2}$, then $a_{1}a_{3}\notin E(G)$ and
$b_{2}a_{1},b_{2}a_{3}$ can not both be edges, which implies
$d(a_{1})+d(a_{3})\leq 8$. So $w(G)\leq\frac{1}{2}(5\cdot 2+4\cdot
2+3+8)=\frac{5}{2}\cdot 7-3$.

If $b^{\prime}_{1}\neq b_{2}$. If $d(a_{1})+d(a_{2})+d(a_{3})=14$, then we can
assume $d(a_{3})=5$ and $d(a_{2})=4$, so the graph is shown in 10$(b)$. Let
$b_{1}b^{\prime}_{1},b_{2}b^{\prime}_{2}\in E(G)$, and
$B^{\prime}=B+b^{\prime}_{1}+b^{\prime}_{2}$. We have
$w(B^{\prime})\leq\frac{1}{2}(5\cdot 4+4\cdot 2+3\cdot 3)=\frac{5}{2}\cdot
9-4$. Hence we may assume that $d(a_{1})+d(a_{2})+d(a_{3})\leq 13$. Let
$B^{\prime}=B+b^{\prime}_{1}$. It is obtained that
$w(B^{\prime})\leq\frac{1}{2}(5\cdot 2+4+3\cdot 2+13)=\frac{5}{2}\cdot
8-\frac{7}{2}$.

(III). $d(b_{1})=d(b_{2})=4$.

There exist edges $b_{1}b^{\prime}_{1},b_{2}b^{\prime}_{2}\in E(G)$, as shown
in Figure 11.

(i). $b^{\prime}_{1}=b_{2}$ and $b^{\prime}_{2}=b_{1}$.

Note that $a_{1}a_{3}\notin E(G)$, which implies $d(a_{3})=4$.

Assume that $d(a_{1})=5$. We have $a_{1}a_{2}\in E(G)$. If $d(a_{2})=4$, there
exists an edge $a_{2}a^{\prime}_{2}$. Then $a_{2}a^{\prime}_{2}$ is a cut edge
and $V(G)\geq 8$. We will show later that if $G$ contains this subgraph, then
$e(G)\leq\frac{5}{2}n-5$. If $d(a_{2})=3$. Then $B$ is a connected component
on $7$ vertices with $15$ edges.

Hence $d(a_{1})=4$. Then $a_{1}a_{2}\notin E(G)$, which implies $d(a_{2})\leq
3$. It follows $w(B)\leq\frac{29}{2}=\frac{5}{2}\cdot 7-3$.

(ii). $b^{\prime}_{1}=b^{\prime}_{2}$.

Similarly, we have $a_{1}a_{3}\notin E(G)$ and $d(a_{3})=4$.

If $d(a_{2})=4$, there exists an edge $a_{2}a^{\prime}_{2}$ with
$d(a^{\prime}_{2})\leq 3$. Let $B=B+a^{\prime}_{2}+b^{\prime}_{1}$. It follows
$w(B^{\prime})\leq\frac{1}{2}(5\cdot 3+4\cdot 4+3\cdot 2)=\frac{5}{2}\cdot
9-4$.

If $d(a_{2})\leq 3$, let $B=B+b^{\prime}_{1}$. It follows
$w(B^{\prime})\leq\frac{1}{2}(5\cdot 3+4\cdot 3+3\cdot 2)=\frac{5}{2}\cdot
8-\frac{7}{2}$.

(iii). $b^{\prime}_{1}\neq b^{\prime}_{2}$.

There are two possible planar embeddings, as show in Figure 11.

For the first planar embedding, $b_{1},b_{2}$ are in a different region from
$a_{2}$. We will show that $d(a_{1})+d(a_{2})+d(a_{3})\leq 13$. In fact, if
$d(a_{3})=5$, then $a_{1}a_{3}\in E(G)$ and $a_{1}a_{2}\notin E(G)$, which
means that $d(a_{2})\leq 3$. Let $B^{\prime}=B+b^{\prime}_{1}+b^{\prime}_{2}$.
Then $w(B^{\prime})\leq\frac{1}{2}(5\cdot 2+4\cdot 2+3\cdot
2+13)=\frac{5}{2}\cdot 9-4$.

For the second planar embedding, $a_{2},b_{2}$ are in a different region from
$b_{1}$. It is easy to check that $d(a_{1})\leq 5,d(a_{2})\leq 4,d(a_{3})\leq
4$. Let $B^{\prime}=B+b^{\prime}_{1}+b^{\prime}_{2}$. Similarly we have
$w(B^{\prime})\leq\frac{37}{2}=\frac{5}{2}\cdot 9-4$.

Now we prove that if $G$ contains such subgraph in (i), then
$e(G)\leq\frac{5}{2}n-5$. When $n\geq 13$, we have $e(G)\leq
w(B)+\frac{1}{2}\cdot 4(n-7)\leq 2n+\frac{3}{2}\leq\frac{5}{2}n-5$.

Recall that $G$ is connected and $a_{2}a^{\prime}_{2}$ is a cut edge. When
$n\leq 12$, we may assume that there are two components $G_{1},G_{2}$
connected by $a_{2}a^{\prime}_{2}$. Then

$\displaystyle e(G)$ $\displaystyle=e(G_{1})+e(G_{2})+1$
$\displaystyle=e(G_{2})+16$ $\displaystyle\leq 3(n-7)-6+16$
$\displaystyle=3n-11$ $\displaystyle\leq\frac{5}{2}n-5.$

Therefore if $G$ contains this certain type of $5$-$5$ edge,
$e(G)\leq\frac{5}{2}n-5$. ∎

Hence,

$\displaystyle e(G)$ $\displaystyle=w(B)+w(G\backslash B)$
$\displaystyle\leq\frac{5}{2}k-\frac{k-1}{2}+\frac{1}{2}\cdot 4(n-k)$
$\displaystyle\leq 2n+\frac{1}{2}.$

Similarly, we have $e(G)\leq 2n$ when $8\leq n\leq 10$ and
$e(G)\leq\frac{5}{2}n-5$ when $n\geq 11$.

Case 4. $G$ contains a $5$-$4$-$5$ path or $5$-$4^{-}$ star.

Based on the proof in Lemma 2.1, it is easy to see that
$w(B)\leq\frac{5}{2}k-\frac{k-1}{2}$. Similarly, $e(G)$ satisfies the upper
bound.

Note that $2n=\frac{5}{2}n-5$ when $n=10$. Therefore the lemma holds.

##  5 Proof of Lemma 2.3 and Theorem 1.1

In this section, we first prove the Lemma 2.3, and then provide the proof of
upper bound in theorem 1.1.

###### Proof.

Note that if there is a vertex of degree at least $5$, then it must be
contained in some star-block. Since $G$ has a star-block partition
$G=G_{1}+G_{2}$, it follows that

$e(G)=w_{0}(G_{1})+w_{0}(G_{2})\leq w_{0}(G_{1})+\frac{1}{2}\sum\limits_{v\in
V(G_{2})}4=w_{0}(G_{1})+2v(G_{2}).$

It suffices to prove that

$w_{0}(G_{1})\leq\frac{5}{2}v(G_{1})-5.$

Recall that $G_{1}$ has a star-block base $\mathcal{B}$. Let

  * •

$r_{1}\coloneqq\big{|}\\{v\in V(G_{1}):d_{\mathcal{B}}(v)=2\ and\ d(v)\leq
3\\}\big{|}$,

  * •

$r_{2}\coloneqq\big{|}\\{v\in V(G_{1}):d_{\mathcal{B}}(v)=2\ and\
d(v)=4\\}\big{|}$,

  * •

$r_{3}\coloneqq\big{|}\\{v\in V(G_{1}):d_{\mathcal{B}}(v)=3\\}\big{|}$.

Then we have

$\sum\limits_{B\in\mathcal{B}}w_{0}(B)=w_{0}(G_{1})+\frac{3}{2}r_{1}+2r_{2}+3r_{3}.$

Let $t_{0}=|\mathcal{B}_{0}|$, $t_{1}=|\mathcal{B}_{1}|$,
$t_{2}=|\mathcal{B}_{2}|$. By Lemma 2.1, we have

$\displaystyle\sum\limits_{B\in\mathcal{B}}w_{0}(B)$
$\displaystyle=\sum\limits_{B\in\mathcal{B}}(w(B)-\frac{s}{2}-\frac{s^{\prime}}{4}-\mathbf{1}_{B})$
$\displaystyle=\sum\limits_{B\in\mathcal{B}}(w(B))-\sum\limits_{B\in\mathcal{B}}(\frac{s}{2}+\frac{s^{\prime}}{4}+\mathbf{1}_{B})$
$\displaystyle\leq\frac{5}{2}\sum\limits_{B\in\mathcal{B}}v(B)-\frac{5}{2}t_{0}-\frac{5t_{1}}{t}-t_{2}-(r_{1}+\frac{1}{2}r_{2}+\frac{3}{2}r_{3}+t_{2})$
$\displaystyle=\frac{5}{2}(v(G_{1})+r_{1}+r_{2}+2r_{3})-\frac{5}{2}t_{0}-\frac{5t_{1}}{t}-t_{2}-(r_{1}+\frac{1}{2}r_{2}+\frac{3}{2}r_{3}+t_{2}).$

Combining the results, we conclude that

$w_{0}(G_{1})\leq\frac{5}{2}v(G_{1})+(\frac{1}{2}r_{3}-t_{2})-\frac{5}{2}t_{0}-\frac{5t_{1}}{t}-t_{2}.$

Next we will show that $r_{3}\leq 2t_{2}-4$ when $t_{2}\neq 0$. Recall that
$r_{3}$ is the number of shared vertices in $G_{1}$ with
$d_{\mathcal{B}}(v)=3$ and $t_{2}$ is the number of star-blocks in
$\mathcal{B}$ containing such type of vertices. We construct an auxiliary
bipartite graph $(X,Y)$ such that $|X|=r_{3}$ and $|Y|=t_{2}$. Each vertex in
$X$ represents a vertex $v$ in $G_{1}$ with $d_{\mathcal{B}}(v)=3$ and each
vertex in $Y$ represents a star-block in $\mathcal{B}_{2}$. Moreover, the edge
$xy$ means that $x$ is contained in the star-block $y$ for $x\in X$ and $y\in
Y$. It is easy to check that the auxiliary bipartite graph is a planar graph
since $G$ is a planar graph and every vertex in $X$ has degree exactly $3$. So
we have $3r_{3}\leq 2(r_{3}+t_{2})-4$. This implies $r_{3}\leq 2t_{2}-4$.

Hence we have

$\displaystyle
w_{0}(G_{1})\leq\frac{5}{2}v(G_{1})-2\cdot\mathbf{1}_{t_{2}}-\frac{5}{2}t_{0}-\frac{5t_{1}}{t}-t_{2},$

where $\mathbf{1}_{t_{2}}$ is the characteristic function of $t_{2}$.

If $t_{2}>0$, then $t_{2}\geq 3$. It follows that
$w_{0}(G_{1})\leq\frac{5}{2}v(G_{1})-5$. Assume that $t_{2}=0$. If $t_{1}>0$,
then $w_{0}(G_{1})\leq\frac{5}{2}v(G_{1})-5$ since $t_{1}=t_{1}+t_{2}=t$. If
$t_{1}=t_{2}=0$, then $w_{0}(G_{1})\leq\frac{5}{2}v(G_{1})-5$ since
$t_{0}=|\mathcal{B}|\geq 2$.

It should be noted that we did not consider vertices of degree 2 in our proof.
In fact, the calculations for $w(B)$ were based on the assumption that all
vertices in $B$ have degree at least $3$. Assume there exists a $2$-degree
vertex $w$ in $G$. When $d_{\mathcal{B}}(w)=1$, it can be easily verified.
When $d_{\mathcal{B}}(w)=2$, we show that this will not affect our proof
either.

Note that the core of the proof is the following inequality

$w_{0}(G_{1})+\frac{3}{2}r_{1}+2r_{2}+3r_{3}\leq\sum\limits_{B\in\mathcal{B}}(w(B)-\frac{s}{2}-\frac{s^{\prime}}{4}-\mathbf{1}_{B}).$

If there exists a shared vertex $w$ of degree $2$, then the value of
$w_{0}(G_{1})+\frac{3}{2}r_{1}+2r_{2}+3r_{3}$ would be reduced by $1/2$.
However $w(B)$ would be reduced by $1/2$ if $w\in B$. And there are two star-
blocks containing $w$. Thus
$\sum\limits_{B\in\mathcal{B}}(w(B)-\frac{s}{2}-\frac{s^{\prime}}{4}-\mathbf{1}_{B})$
would be reduced by at least $1$. Hence, ignoring each shared vertex of degree
$2$ results in an increase of $\frac{1}{2}$ to the value on the left side of
the inequality, and an increase of $1$ to the right side. This does not affect
our calculation above. ∎

Now we give the proof of Theorem 1.1.

###### Proof.

Given any planar graph $G$ on $n\leq 7$ vertices, there does not exist an
$S_{3,3}$ obviously. Thus $e(G)\leq 3n-6$.

By Lemma 2.2 and Lemma 2.3, it remains to prove the theorem for
$|\mathcal{B}|=0$. This means there is no star-block contained in $G$. Thus we
have $\Delta(G)\leq 4$. Hence $e(G)=\sum\limits_{v\in V(G)}d(v)\leq 2n$. When
$n\geq 10$, we have $e(G)\leq\frac{5}{2}n-5$.

Therefore the proof is completed. ∎

##  6 Construction of Extremal Graphs

In the previous sections, we have shown the upper bound of edges for
$S_{3,3}$-free planar graphs. Now we shall complete it by demonstrating that
this bound is tight.

If $3\leq n\leq 7$, any $n$-vertex maximal planar graph is the extremal graph.
If $n=8,9$, the $4$-regular planar graph is the extremal graph, as shown in
Figure 13$(a)(b)$. Moreover, the extremal graph can also be constructed by
some star-block, as shown in Figure 13$(c)$.

Figure 13: Extremal graphs on $8$ or $9$ vertices.

Next we assume that $n\geq 10$. Let $G=G_{1}+G_{2}$ be a star-block partition,
where $G_{1}$ is the union of star-blocks and all vertices in $G_{2}$ have
degree at most $4$. We also use $t_{0},t_{1},t_{2}$ to denote the size of
$\mathcal{B}_{0},\mathcal{B}_{1},\mathcal{B}_{2}$. Let $t=t_{1}+t_{2}$.

It is known that if $t>0$, the number of edges in extremal graphs must attain
the equality in Lemma 2.3.

If $t_{2}>0$, the equation holds when $t_{0}=t_{1}=|V(G_{2})|=0$ and
$t_{2}=3$. This means all shared vertices are of degree $3$ and there are
exactly $3$ star-blocks. Since each shared vertex is connected with all star-
blocks, the number of shared vertices is at most $2$, according to the
property of planar graph. It can be checked that the possible star-block is
$6$-$5$ edge, or $6$-$4$ edge, or $5$-$5$ edge. However we obtain that $w(B)$
will be reduced strictly here. Thus there does not exist such extremal graph.

Assume that $t_{2}=0$. If $t_{1}>0$, the equation holds when
$t_{0}=|V(G_{2})|=0$ and $t_{2}=2$. This means that there are exactly $2$
star-blocks and all shared vertices are of degree at most $3$. Here we
construct two extremal graphs by combining different star-blocks. The first
extremal graph is obtained by merging a $6$-$6$ edge and a $6$-$5$ edge, as
shown in Figure 14$(a)$. There are $13$ vertices and $27$ edges.

Figure 14: Extremal graphs constructed by merging two star-blocks.

The second extremal graph is constructed by two $k$-$3^{-}$ stars, where
$k=n-2$. When $n$ is even, the graph obtained from two $k$-$3$ stars is the
extremal graph, as shown in Figure 14$(b)$. When $n$ is odd, the extremal
graph is constructed by two $k$-$3^{-}$ stars, where in each star-block, $k-1$
peripheral vertices have degree $3$ and one has degree $2$, as shown in Figure
14$(c)$. Moreover, $e(G)=\lfloor 5n/2\rfloor-5$.

Figure 15: Extremal graphs on $14$ vertices and $30$ edges.

Finally, we assume that $t_{1}=t_{2}=0$. If $t_{0}=1$, the graph in Figure
3$(b)$ is an example on $7$ vertices. It remains to discuss the case when
$t_{0}=2$. Here we give an extremal graph by connecting two star-blocks, as
shown in Figure 15. Furthermore, if $n=13,14$, there are extremal graphs that
are disconnected. Let $G_{1},G_{2}$ be the two connected components, where
$G_{1}$ is the $7$-vertex planar triangulation and $G_{2}$ is the $6$-vertex
or $7$-vertex planar triangulation. It is noticed that there are five
$7$-vertex maximal planar graphs, each contains some star-block discussed
above.

##  7 Remark

If someone can use a computer to enumerate and verifies Conjecture 1.1 holds
for $n\leq 27$, the proof would be much simpler by slightly modifying the
induction-based approach of Ghosh, Győri, Paulos and Xiao [10].

Now we can assume that Conjecture 1.1 holds for $n\leq m-1$, where $m\geq 28$,
and let $G$ be an $m$-vertex $S_{3,3}$-free graph. By inductive hypothesis we
may assume that $\delta(G)\geq 3$ and $G$ contains no $3$-$3$ edge. Using the
same proof of Ghosh, Győri, Paulos and Xiao [10], we can assume that $G$
contains no $6$-$6$ edge, $6$-$5$ edge, $6$-$4$ edge. Since $G$ is
$S_{3,3}$-free, $G$ contains no $7^{+}$-$4^{+}$ edge. Now $G$ is shown in
Figure 16.

Figure 16: $G$ can be partitioned into three parts $A$, $B$, $C$.

Let $m_{3},m_{4},m_{5}$ be the number of vertices of degree $3$, $4$, $5$,
respectively. Let $m_{6}$ be the number of vertices of degree at least $6$.
Then we have $m_{3}+m_{4}+m_{5}+m_{6}=m$. Let $x$ be the number of edges
between $A$ and $B$.

If $m_{6}\geq 2$,

$\displaystyle e(G)$ $\displaystyle=3m_{3}+\frac{1}{2}(4m_{4}+5m_{5}-x)$
$\displaystyle=\frac{5}{2}(m-m_{6})+\frac{1}{2}m_{3}-\frac{1}{2}m_{4}-\frac{1}{2}x$
$\displaystyle\leq\frac{5}{2}m-\frac{3}{2}m_{6}-\frac{1}{2}m_{4}-2$
$\displaystyle\leq\frac{5}{2}m-5,$

where the first inequality follows from counting the number of edges between
$B$ and $C$, which is $3m_{3}-x\leq 2(m_{3}+m_{6})-4$, where the right side is
the maximum number of edges in the planar bipartite graph induced by $B$ and
$C$.

If $m_{6}=1$,

$\displaystyle e(G)$ $\displaystyle=3m_{3}+\frac{1}{2}(4m_{4}+5m_{5}-x)$
$\displaystyle=\frac{5}{2}(m-m_{6})+\frac{1}{2}m_{3}-\frac{1}{2}m_{4}-\frac{1}{2}x$
$\displaystyle\leq\frac{5}{2}m-\frac{1}{2}m_{3}-\frac{1}{2}m_{4}-\frac{5}{2},$

where the last inequality follows from counting the number of edges between
$B$ and $C$, which is $3m_{3}-x\leq m_{3}$. If $m_{3}+m_{4}\geq 5$ then we are
done, otherwise $m_{3}+m_{4}\leq 4$, which means $|B|=m_{3}\leq 4$. However,
the vertex in $C$ is adjacent to at least $6$ vertices in $B$, a
contradiction.

If $m_{6}=0$, it is easy to check that every $5$-vertex is adjacent to at
least $2$ vertices of degree at most $3$ since $G$ is $S_{3,3}$-free. So we
have $2m_{5}\leq 3m_{3}+4m_{4}$. Then
$e(G)=\frac{1}{2}(3m_{3}+4m_{4}+5m_{5})=\frac{5}{2}m-\frac{1}{2}m_{4}-m_{3}$.
If $2m_{3}+m_{4}\geq 10$ then we are done, so we may assume that
$2m_{3}+m_{4}\leq 9$. Now $m=m_{3}+m_{4}+m_{5}\leq\frac{5}{2}m_{3}+3m_{4}\leq
27-\frac{7}{2}m_{3}\leq 27$, a contradiction.

Therefore the proof is completed.

## Acknowledgments

Tong Li and Guiying Yan are partially supported by National Natural Science
Foundation of China (Grant No. 12301459, 11631014).

## References

  * [1] B. Bollobás.  Modern graph theory.  In Graduate Texts in Mathematics, 2002.
  * [2] D. W. Cranston, B. Lidický, X. Liu, and A. Shantanam.  Planar Turán numbers of cycles: a counterexample.  Electronic Journal of Combinatorics, 29(3):Paper No. 3.31, 10, 08 2022.
  * [3] C. Dowden.  Extremal ${C}_{4}$-free/${C}_{5}$-free planar graphs.  Journal of Graph Theory, 83(3):213–230, 2016.
  * [4] L. Du, B. Wang, and M. Zhai.  Planar Turán numbers on short cycles of consecutive lengths.  Bulletin of the Iranian Mathematical Society, 48(5):2395–2405, 2022.
  * [5] P. Erdős and A. H. Stone.  On the structure of linear graphs.  Bulletin of the American Mathematical Society, 52(12):1087–1091, 1946.
  * [6] L. Fang, H. Lin, and Y. Shi.  Extremal spectral results of planar graphs without vertex-disjoint cycles, arxiv: 2304.06942, 2023.
  * [7] L. Fang, B. Wang, and M. Zhai.  Planar Turán number of intersecting triangles.  Discrete Mathematics, 345(5):Paper No. 112794, 10, 2022.
  * [8] R. Faudree and R. Schelp.  Path Ramsey numbers in multicolorings.  Journal of Combinatorial Theory, Series B, 19(2):150–160, 1975.
  * [9] D. Ghosh, E. Győri, R. R. Martin, A. Paulos, and C. Xiao.  Planar Turán number of the 6-cycle.  SIAM Journal on Discrete Mathematics, 36(3):2028–2050, 2022.
  * [10] D. Ghosh, E. Győri, A. Paulos, and C. Xiao.  Planar Turán number of double stars, arxiv: 2110.10515, 2022.
  * [11] D. Ghosh, E. Győri, A. Paulos, C. Xiao, and O. Zamora.  Planar Turán number of the $\theta_{6}$, arxiv: 2006.00994, 2023.
  * [12] E. Győri, A. Li, and R. Zhou.  The planar Turán number of the seven-cycle, arxiv: 2307.06909, 2023.
  * [13] E. Győri, A. Li, and R. Zhou.  The planar Turán number of $\\{K_{4},C_{5}\\}$ and $\\{K_{4},C_{6}\\}$, arxiv: 2308.09185, 2023.
  * [14] E. Győri, K. Varga, and X. Zhu.  A new construction for planar Turán number of cycle, arXiv: 2304.05584, 2023.
  * [15] E. Győri, X. Wang, and Z. Zheng.  Extremal planar graphs with no cycles of particular lengths, arxiv: 2208.13477, 2022.
  * [16] Y. Lan and Y. Shi.  Planar Turán numbers of short paths.  Graphs and Combinatorics, 35(5):1035–1049, 2019.
  * [17] Y. Lan, Y. Shi, and Z.-X. Song.  Extremal $H$-free planar graphs.  Electronic Journal of Combinatorics, 26(2):No. 2.11, 17, 2019.
  * [18] Y. Lan, Y. Shi, and Z.-X. Song.  Extremal theta-free planar graphs.  Discrete Mathematics, 342(12):111610, 8, 2019.
  * [19] Y. Lan, Y. Shi, and Z.-X. Song.  Planar Turán numbers of cubic graphs and disjoint union of cycles.  Graphs and Combinatorics, 40(2):No. 28, 2024.
  * [20] Y. Lan and Z.-X. Song.  An improved lower bound for the planar Turán number of cycles, arxiv: 2209.01312, 2022.
  * [21] P. Li.  Planar Turán number of the disjoint union of cycles.  Discrete Applied Mathematics, 342:260–274, 2024.
  * [22] R. Shi, Z. Walsh, and X. Yu.  Planar Turán number of the 7-cycle, arxiv: 2306.13594, 2023.
  * [23] R. Shi, Z. Walsh, and X. Yu.  Dense circuit graphs and the planar Turán number of a cycle, arxiv: 2310.06631, 2023.
  * [24] P. Turán.  On an extremal problem in graph theory.  Matematikai és Fizikai Lapok, 48:436–452, 1941.
  * [25] X. Xu, Y. Hu, and X. Zhang.  An improved upper bound for planar Turán number of double star ${S}_{2,5}$, arxiv: 2404.11060, 2024.
  * [26] M. Zhai and M. Liu.  Extremal problems on planar graphs without k edge-disjoint cycles, arxiv: 2207.09681, 2022.


# Nucleon off-shell structure and the free neutron valence structure from A=3
inclusive electron scattering measurements

E.P. Segarra  Massachusetts Institute of Technology, Cambridge, Massachusetts
02139, USA  J.R. Pybus  Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139, USA  F. Hauenstein  Massachusetts Institute of
Technology, Cambridge, Massachusetts 02139, USA  Old Dominion University,
Norfolk, Virginia 23529, USA  T. Kutz  Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA  School of Physics and Astronomy, Tel Aviv
University, Tel Aviv 69978, Israel  D. Higinbotham  Thomas Jefferson National
Accelerator Facility, Newport News, Virginia 23606, USA  G.A. Miller  Fermi
National Accelerator Laboratory, Batavia, IL 60510, USA  E. Piasetzky  School
of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel  A.
Schmidt  George Washington University, Washington, D.C., 20052, USA  M.
Strikman  Pennsylvania State University, University Park, PA, 16802, USA  L.B.
Weinstein  Old Dominion University, Norfolk, Virginia 23529, USA  O. Hen
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
<EMAIL_ADDRESS>
###### Abstract

Understanding the differences between the distribution of quarks bound in
protons and neutrons is key for constraining the mechanisms of SU(6) spin-
flavor symmetry breaking in Quantum Chromodynamics (QCD). While vast amounts
of proton structure measurements were done, data on the structure of the
neutron is much more spars as experiments typically extract the structure of
neutrons from measurements of light atomic nuclei using model-dependent
corrections for nuclear effects. Recently the MARATHON collaboration performed
such an extraction by measuring inclusive deep-inelastic electron-scattering
on helium-3 and tritium mirror nuclei where nuclear effects are expected to be
similar and thus be suppressed in the helium-3 to tritium ratio. Here we
evaluate the model dependence of this extraction by examining a wide range of
models including the effect of using instant-form and light-cone nuclear wave
functions and several different parameterizations of nucleon modification
effects, including those with and without isospin dependence. We find that,
while the data cannot differentiate among the different models of nuclear
structure and nucleon modification, they consistently prefer a neutron-to-
proton structure function ratio of at $x_{B}\rightarrow 1$ of $\sim 0.4$ with
a typical uncertainty ($1\sigma$) of $\sim 0.05$ and $\sim 0.10$ for isospin-
independent and isospin-dependent modification models, respectively. While
strongly favoring SU(6) symmetry breaking models based on perturbative QCD and
the Schwinger–Dyson equation calculation, the MARATHON data do not completely
rule out the scalar di-quark models if an isospin-dependent modification
exist.

Detailed studies of the distribution of quarks bound in nucleons provides
crucial insight into the theory of Quantum Chromodynamics (QCD), thereby
improving our understanding of emergent QCD phenomena such as baryon
structure, masses, and magnetic moments, the structure of matter, and the
origin of visible mass in the universe Bashir  _et al._ (2012); Roberts  _et
al._ (2013). A typical measurement used for such studies is inclusive lepton
deep-inelastic scattering (DIS) Bjorken and Paschos (1969).

Comparing the distribution of high-momentum quarks in the proton and neutron
is especially sensitive to models of SU(6) spin-flavor symmetry breaking
mechanism in QCD, and thus have far reaching implications to our understanding
of the fundamental structure of matter Roberts  _et al._ (2013). While vast
amounts of proton DIS data exist, the lack of a free neutron target prevents
equivalent measurements of the neutron DIS, challenging our ability to perform
a direct test of QCD symmetry breaking mechanisms. Instead, the free neutron
structure is extracted from measurements that are probing protons and neutrons
bound in atomic nuclei, applying model-dependent theoretical corrections for
nuclear effects such as binding, nucleon-motion, and nucleon structure
modification Segarra  _et al._ (2020a); Dulat  _et al._ (2016); Accardi  _et
al._ (2016); Arrington  _et al._ (2012). The latter are minimized, but not
cancelled, by focusing on the lightest atomic nuclei with 2 and three
nucleons, where exact nuclear wave functions can be calculated and the effects
of nucleon binding and Fermi motion and nucleon modification are generally
small.

Here we use a wide model phase-space to analyze the recent Helium-3 and
Tritium mirror-nuclei DIS measurements of the MARATHON collaboration Abrams
_et al._ (2021) to examine the model systematic uncertainty associated with
the constrain in places on the distribution of valance quarks in the free
proton and neutron. Significant emphasis is placed on the correlation between
the free neutron structure and bound nucleon modification which prevented such
extractions using previous Deuterium and Helium-3 measurements but is
mitigated by the use of Tritium and assumptions of isospin symmetry.

Figure 1:  Fit results for an isospin-independent off-shell nucleon
modification function. Left: results for a constant-in-$x$ off-shell function.
Right: results for a linear-in-$x$ off-shell function. In each block, the
three left panels show the best-fit results for the BONuS (top) Griffioen  _et
al._ (2015), Seely (middle) Seely  _et al._ (2009), and MARATHON (bottom)
Abrams  _et al._ (2021) data. The upper right panels shows a two-dimensional
90% confidence region for $F_{2}^{n}/F_{2}^{p}\big{|}_{x_{B}\to 1}$ and one
off-shell parameter. The lower right panels shows the best-fit result
translated into a probability distribution function (PDF) for the extracted
value of $F_{2}^{n}/F_{2}^{p}\big{|}_{x_{B}\to 1}$ expected if the fit
procedure were repeated on an equivalent independent data set. Also, values of
$F_{2}^{n}/F_{2}^{p}\big{|}_{x_{B}\to 1}$ predicted by various models
(translated from $d/u\big{|}_{x\to 1}$ using the simple parton model) are
indicated by dotted vertical lines. As described in the text, blue curves show
results when excluding MARATHON data, while red curves include MARATHON data
in the fit. Solid curves use the spectral-function formulation for
$\rho_{N}^{A}(\alpha,v)$, while dashed-curves use the light-cone formulation.

The cross-section for lepton DIS off stationary nucleon targets at fixed
momentum transfer squared $Q^{2}=|\vec{q}|^{2}-\nu^{2}$ and Bjorken scaling
parameter $x_{B}=Q^{2}/2m_{N}\nu$ is directly proportional to the inelastic
structure function of the target nucleon, $F_{2}(Q^{2},x_{B})$ in the limit
$F_{2}=2xF_{1}$. Here $q=(\vec{q},\nu)$ is the four-momentum transfer from the
electron to the target carried by the exchange of a virtual photon, and
$m_{N}$ is the nucleon mass. The parameter $x_{B}$ coincides, in the infinite-
momentum frame and $Q^{2}\rightarrow\infty$, with the momentum fraction, $x$,
of a parton struck in the scattering reaction.

In the simple quark-parton model, $F_{2}(Q^{2},x_{B})$ is the linear
combination of individual quark flavor distributions (up, down, strange…),
each weighted by the square of its electromagnetic charge. Therefore,
$F_{2}(Q^{2},x_{B})$ structure function extractions from lepton-nucleon DIS
measurements contain valuable information on the internal structure of the
nucleon.

To test the finer predictions of QCD, it is highly desirable to separate
$F_{2}(Q^{2},x_{B})$ into the individual quark flavor distributions.
Specifically, the ratio of the proton’s down to up quark distributions, $d/u$,
in the $x\rightarrow 1$ limit has been shown to be extremely sensitive to the
SU(6) spin-flavor symmetry breaking mechanism in QCD Roberts  _et al._ (2013).
SU(6) is a broken symmetry in QCD, as is clear from the mass difference
between the nucleon and $\Delta$-baryon, but if it were preserved, then the
proton’s $d/u$ ratio at $x=1$ would be 1/2 Close (1979); the proton has two
valence up quarks, but only a single valence down quark. Specific models of
the symmetry breaking mechanism predict a range of end points for $d/u$. A
scalar di-quark picture predicts $d/u\big{|}_{x\to 1}=0$ Close (1973); Carlitz
(1975), perturbative QCD predicts $d/u\big{|}_{x\to 1}=1/5$ Farrar and Jackson
(1975), and modern calculation using the Dyson-Schwinger equation predict a
range of end points between 0.18 and 0.27 Roberts  _et al._ (2013). A precise
experimental determination would therefore offer valuable insight into how
this symmetry is broken in QCD.

Figure 2:  Fit results for an isospin-dependent off-shell nucleon modification
function. The panels are arranged as in Fig. 1.

Performing such flavor decomposition is a challenging task, which is often met
by using isospin symmetry in comparing data from free proton and bound neutron
measurements of light-nuclei, with appropriate theoretical corrections to
obtain the free neutron structure. While advances in nuclear structure theory
have greatly improved the accuracy of nucleon motion and binding effects,
there is still the problem of the largely unconstrained effects of the
modification of the quark distributions in bound nucleons Hen  _et al._
(2017). This bound structure modification is known as the EMC effect, and is
the main limitation in extracting the free neutron structure function from
inclusive DIS measurements of light nuclei.

This fundamental limitation is currently being addressed by several novel
measurements, such as those of the BONuS Baillie  _et al._ (2012); Tkachenko
_et al._ (2014); Bueltmann  _et al._ (2010) and MARATHON Petratos  _et al._
(2010); Abrams  _et al._ (2021) experiments. BONuS measures tagged-DIS (TDIS)
reactions to identify scattering events from nearly free neutrons in
deuterium. MARATHON measures the ratio of the inclusive DIS cross-section of
helium-3 (3He) and tritium (3H) mirror nuclei, where nuclear effects are
expected to mostly cancel in the ratio due to the similarity in the nuclear
structure of the two nuclei. Recently, the MARATHON collaboration published
its first results for the 3He/3H DIS cross-section ratio and, using the
calculations of Kulagin and Petti (KP) Kulagin and Petti (tion, 2010), applied
model-dependent corrections to extract the free neutron to proton structure
function ratio $F_{2}^{n}(Q^{2},x_{B})/F_{2}^{p}(Q^{2},x_{B})$ Abrams  _et
al._ (2021).

Inspired by previous studies that indicate a potential systematic model
uncertainty of up to $20\%$ in this extraction Segarra  _et al._ (2020a, b),
we examine the impact of the variation across the space of models on the
accuracy of the $F_{2}^{n}(Q^{2},x_{B})/F_{2}^{p}(Q^{2},x_{B})$ extraction
from the MARATHON data. Our study includes both instant-form and light-cone
nuclear wave functions as well as a broad range of nuclear modification models
that all reproduce existing DIS data from light nuclei Segarra  _et al._
(2020b).

Within the standard nuclear convolution approximation, the measured inelastic
structure function of a nuclear target, $F_{2}^{A}(x_{B})$, is related to the
free proton and neutron structure functions by Frankfurt and Strikman (1985,
1987, 1988); Akulinichev  _et al._ (1985); Dunne and Thomas (1985):

$\begin{split}&F_{2}^{A}(x_{B})=\\\
&\frac{1}{A}\int_{x_{B}}^{A}\frac{d\alpha}{\alpha}\int_{-\infty}^{0}dv\>F_{2}^{p}\big{(}\tilde{x}\big{)}\left[Z\tilde{\rho}_{p}^{A}(\alpha,v)+N\tilde{\rho}_{n}^{A}(\alpha,v)\frac{F_{2}^{n}(\tilde{x})}{F_{2}^{p}(\tilde{x})}\right]\\\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\times\Big{(}1+v\
f^{\text{off}}(\tilde{x})\Big{)},\end{split}$ (1)

where $Z$ and $N$ are respectively the proton and neutron numbers of the
target nucleus, $A=Z+N$, $\alpha$ is the nucleon light-cone momentum fraction,
$v$ is the fractional virtuality of the bound nucleon defined by
$v=(E^{2}-|{\bf{p}}|^{2}-m_{N}^{2})/m_{N}^{2}$,
$\tilde{\rho}_{N}^{A}(\alpha,v)$ are the nucleon ($N$ = $p$ or $n$) light-cone
momentum and virtuality distributions in nucleus $A$, $F_{2}^{N}(\tilde{x})$
are the free nucleon structure functions, $\tilde{x}=\frac{Q^{2}}{2p\cdot q}$,
where and $p$ is the four-momentum of the struck off-shell nucleon, and the
function $f^{\text{off}}(\tilde{x})$ is the bound nucleon off-shell
modification function. When working in coordinates where $\hat{z}$ is opposite
the direction of $\vec{q}$, $\alpha=A(E+p_{z})/m_{A}$. For brevity we omit the
explicit $Q^{2}$ dependences of $F_{2}^{p}$, $F_{2}^{n}$, and $F_{2}^{A}$, but
note that they are evaluated at the same $Q^{2}$ value.

This convolution combines nuclear wave function input in the form of the
nucleon momentum and virtuality distributions from relatively well-known
nuclear structure calculations with partonic input in the form of
nuclear/nucleon structure and off-shell modification functions. While the
nuclear and free proton structure functions $F_{2}^{A}$ and $F_{2}^{p}$ are
directly measured, both the neutron-to-proton structure function ratio
${F_{2}^{n}}/{F_{2}^{p}}$ and bound nucleon off-shell modification function
$f^{\text{off}}$ are not.

As can be seen by Eq. 1, for a given nucleus $A$, one can shift strength
between ${F_{2}^{n}}/{F_{2}^{p}}$ and $f^{\text{off}}$ while still reproducing
$F_{2}^{A}$ data. Therefore, ${F_{2}^{n}}/{F_{2}^{p}}$ and off-shell nucleon
modification are correlated when studying a single asymmetric nucleus and/or
several symmetric nuclei.

This situation is different for case of $A=3$ mirror nuclei where, by
exploiting isospin symmetry, i.e.
$\tilde{\rho}_{p}^{{}^{3}\text{He}}(\alpha,v)=\tilde{\rho}_{n}^{{}^{3}\text{H}}(\alpha,v)$
and
$\tilde{\rho}_{n}^{{}^{3}\text{He}}(\alpha,v)=\tilde{\rho}_{p}^{{}^{3}\text{H}}(\alpha,v)$,
and by assuming $f^{\text{off}}$ is the same for both nuclei, the convolution
equations become

$\begin{split}&F_{2}^{{}^{3}\text{He}}(x_{B})=\\\
&\frac{1}{3}\int_{x_{B}}^{3}\frac{d\alpha}{\alpha}\int_{-\infty}^{0}dv\>F_{2}^{p}\big{(}\tilde{x}\big{)}\left[2\tilde{\rho}_{p}^{{}^{3}\text{He}}(\alpha,v)+\tilde{\rho}_{n}^{{}^{3}\text{He}}(\alpha,v)\frac{F_{2}^{n}(\tilde{x})}{F_{2}^{p}(\tilde{x})}\right]\\\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\times\Big{(}1+v\
f^{\text{off}}(\tilde{x})\Big{)},\\\ &F_{2}^{{}^{3}\text{H}}(x_{B})=\\\
&\frac{1}{3}\int_{x_{B}}^{3}\frac{d\alpha}{\alpha}\int_{-\infty}^{0}dv\>F_{2}^{p}\big{(}\tilde{x}\big{)}\left[\tilde{\rho}_{n}^{{}^{3}\text{He}}(\alpha,v)+2\tilde{\rho}_{p}^{{}^{3}\text{He}}(\alpha,v)\frac{F_{2}^{n}(\tilde{x})}{F_{2}^{p}(\tilde{x})}\right]\\\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\times\Big{(}1+v\
f^{\text{off}}(\tilde{x})\Big{)},\\\ \end{split}$ (2)

and the degeneracy between ${F_{2}^{n}}/{F_{2}^{p}}$ and $f^{\text{off}}$ is
broken. Therefore, data from mirror nuclei should improve constraints on
${F_{2}^{n}}/{F_{2}^{p}}$, although the exact results obtained can still
depend on the assumed model $f^{\text{off}}$. Indeed, while previous studies
show that inclusive DIS data of deuterium and helium-3 can be explained using
a wide range of functional forms of $f^{\text{off}}$ Segarra  _et al._
(2020b), the MARATHON collaboration’s extraction of ${F_{2}^{n}}/{F_{2}^{p}}$
used only one model for $f^{\text{off}}$ by KP.

Figure 3:  Fit results for the CJ and KP off-shell nucleon modification
function. The left and middle columns shows the best-fit results compared to
the BONuS (top), Seely (middle), and MARATHON (bottom) data for the CJ and KP
off-shell functions respectively. The right column shows the best-fit result
translated into a probability distribution function (PDF) for
$F_{2}^{n}/F_{2}^{p}\big{|}_{x_{B}\to 1}$, exactly as in Figs. 1 and 2.

Here we study the model dependence of the
${F_{2}^{n}}/{F_{2}^{p}}\big{|}_{x_{B}\to 1}$ extraction from the MARATHON
3He/3H data. First, we use both light-cone and spectral function approaches
for the calculation of $\tilde{\rho}_{N}^{{}^{3}\text{He}}(\alpha,v)$. Both
approaches are detailed in Ref. Segarra  _et al._ (2020b), but we emphasize
that $\tilde{\rho}_{N}^{A}(\alpha,v)$ satisfy baryon and momentum sum rules in
the light-cone approach while the momentum sum rule is slightly violated in
the spectral function approach. Second, we examine the impact of different
off-shell modification models. Note from Eq. LABEL:Eq:F2_A3 that dependence on
nucleon virtuality is built into the convolution. We explore various
$\tilde{x}$ dependencies by applying the following set of $f^{\text{off}}$
functions :

$\displaystyle f^{\text{off}}_{\text{const}}(\tilde{x})$ $\displaystyle=$
$\displaystyle C,$ (3) $\displaystyle
f^{\text{off},N}_{\text{const\;iso}}(\tilde{x})$ $\displaystyle=$
$\displaystyle C_{N},$ (4) $\displaystyle
f^{\text{off}}_{\text{lin}\;x}(\tilde{x})$ $\displaystyle=$ $\displaystyle
a+b\cdot\tilde{x}$ (5) $\displaystyle f^{\text{off}}_{KP,\;CJ}(\tilde{x})$
$\displaystyle=$ $\displaystyle
C(x_{0}-\tilde{x})(x_{1}-\tilde{x})(1+x_{0}-\tilde{x})$ (6)

The first two models assume modification that is independent of $\tilde{x}$,
and is either the same ($f^{\text{off}}_{\text{const}}$) or different
($f^{\text{off},N}_{\text{const\;iso}}$) for neutrons and protons. The third
model assumes a linear dependence on $\tilde{x}$. The fourth uses a more
complex function form whose free parameters ($C$, $x_{0}$, and $x_{1}$) were
previously determined by the global analyses of Ref. Kulagin and Petti (2006)
(KP) and Accardi  _et al._ (2016) (CJ) and are held fixed in his work.

Using these $f^{\text{off}}$ parameterizations, we performed a simultaneous
fit of Eq. 1 to the ${F_{2}^{d}}/({F_{2}^{p}}+{F_{2}^{n}})$ and
${F_{2}^{{}^{3}\text{He}}}/{F_{2}^{d}}$ data by BONuS Griffioen  _et al._
(2015) and Seely Seely  _et al._ (2009) respectively, and examined the
correlation between the off-shell modification parameters and
${F_{2}^{n}}/{F_{2}^{p}}\big{|}_{x_{B}\to 1}$. We then repeated the fits
including the MARATHON ${F_{2}^{{}^{3}\text{He}}}/{F_{2}^{{}^{3}\text{H}}}$
data and examined the impact on the correlations.

In our fits, we used the $\frac{F_{2}^{n}(\tilde{x})}{F_{2}^{p}(\tilde{x})}$
parametrization of Ref Segarra  _et al._ (2020b):

$\frac{F_{2}^{n}(\tilde{x})}{F_{2}^{p}(\tilde{x})}\equiv
R_{np}(\tilde{x})=a_{np}\left(1-\tilde{x}\right)^{b_{np}}+c_{np},$ (7)

which is both functionally flexible and has a single fit parameter $c_{np}$
that describes ${F_{2}^{n}}/{F_{2}^{p}}\big{|}_{x_{B}\to 1}$. This
parameterization neglects the weak $Q^{2}$-dependence of
$F_{2}^{n}/F_{2}^{p}$. We do include the full $Q^{2}$-dependencee of
$F_{2}^{p}$, for which we use the results of the GD11-P global analyis
Airapetian  _et al._ (2011).

Figure 1 shows the results for fitting the data using the
$f^{\text{off}}_{\text{const}}$ and $f^{\text{off}}_{\text{lin}\;x}$ models.
In all figures, results obtained using light-cone and spectral function
formulations of the nuclear wave function are shown in dashed and solid lines,
respectively. Fits with and without the MARATHON data are shown in red and
blue, respectively. All confidence region contours are shown at the $90\%$
confidence level.

As can be seen, without the MARATHON data a very broad correlation exists
between the extracted ${F_{2}^{n}}/{F_{2}^{p}}\big{|}_{x_{B}\to 1}$ and the
off-shell modification. This results in a wide distribution for
${F_{2}^{n}}/{F_{2}^{p}}\big{|}_{x_{B}\to 1}$ that is centered near the SU(6)
symmetry prediction of $2/3$. Adding the MARATHON
${F_{2}^{{}^{3}\text{He}}}/{F_{2}^{{}^{3}\text{H}}}$ data significantly limits
this correlation and results in ${F_{2}^{n}}/{F_{2}^{p}}\big{|}_{x_{B}\to 1}$
= $0.43\pm 0.04(1\sigma)$ for the constant and linear modification models.
These values are consistent with the pQCD and DSE predictions and reject the
scalar diquark prediction by $4.5$ standard deviations. In both cases the
light-cone and spectral function nuclear wave function formulations have
minimal impact on ${F_{2}^{n}}/{F_{2}^{p}}\big{|}_{x_{B}\to 1}$.

Next, we examine the case of isospin-dependent modification of protons and
neutrons (Fig. 2). In this case, without the MARATHON data
${F_{2}^{n}}/{F_{2}^{p}}\big{|}_{x_{B}\to 1}$ is essentially unconstrained.
With the MARATHON data, it is it constrained to $0.40\pm 0.07(1\sigma)$ with
constant-in-$x$ modification, and to $0.43\pm 0.10(1\sigma)$ and $0.49\pm
0.07(1\sigma)$ for the SF and LC approaches, respectively, with linear
modification. These constraints are in good agreement with the isospin
independent results of Fig 1, but with larger uncertainties that reject the
scalar diquark assumption by only 2–3 standard deviations.

We also show in Fig. 3 the results of our fit using the CJ and KP
$f^{\text{off}}$ parameterizations. We note again that we did not re-fit their
parameters for $f^{\text{off}}$. While their functional form for
$f^{\text{off}}$ is very different from the simple constant and linear forms
discussed above, the resulting constraint on
${F_{2}^{n}}/{F_{2}^{p}}\big{|}_{x_{B}\to 1}$ is similar with values of
$0.42\pm 0.04(1\sigma)$ and $0.43\pm 0.04(1\sigma)$ for CJ and KP
respectively.

Looking at the quality of the fits, we note that all models do a very good job
describing the MARATHON ${F_{2}^{{}^{3}\text{He}}}/{F_{2}^{{}^{3}\text{H}}}$
data. The Seely ${F_{2}^{{}^{3}\text{He}}}/{F_{2}^{d}}$ data is well
reproduced up to $x_{B}\sim 0.8$, above which the invariant mass, $W$, of the
kinematics drops below $1.4$ GeV. The situation is similar for the BONuS data,
which is well reproduced by all calculations except by the KP $f^{\text{off}}$
parametrization at high-$x_{B}$, where the kinematics are also at very low
$W$.

Another interesting feature we observe in Figs. 1 and 2 is that while the
addition of the MARATHON ${F_{2}^{{}^{3}\text{He}}}/{F_{2}^{{}^{3}\text{H}}}$
data significantly constrains ${F_{2}^{n}}/{F_{2}^{p}}\big{|}_{x_{B}\to 1}$,
it is less effective in constraining the off-shell modification parameters,
whose likelihood distributions still span a wide range and have non-negligible
sensitivity to the nuclear wave function formulation (i.e. spectral function
vs. light-cone). This reaffirms the conclusions of Ref. Segarra  _et al._
(2020b) that off-shell modification cannot be well constrained using inclusive
DIS data, even when studying nuclei with different proton-neutron asymmetries.

Last we present the extracted $F_{2}^{n}/F_{2}^{p}$ ratio as a function of
$x_{B}$ in Fig. 4 for both the isospin-independent and isospin-dependent off-
shell modifications. As expected, with an isospin-dependent modification
assumption, $F_{2}^{n}/F_{2}^{p}$ is less constrained, though still consistent
with the pQCD and DSE predictions yet cannot strongly reject SU(6) or scalar
diquark predictions. Since we do not currently consider available data
constraining $F_{2}^{d}$ Arneodo  _et al._ (1995) at very low-$x_{B}$, our
model uncertainty is large in the low-$x_{B}$ region ($\sim 0.1-0.3$),
especially for the isospin-dependent assumption. We also note that the $x_{B}$
value we are sensitive to in $F_{2}^{n}/F_{2}^{p}$ is smeared out due to the
convolution process of Eq. 1 (i.e. $\tilde{x}\leq x_{B}$). However, as seen in
Fig. 4, the saturation seen around $x_{B}\sim 0.7$ gives confidence in our
extrapolation to $x_{B}\rightarrow 1$.

Figure 4:  $R_{np}(x_{B})$ extracted with MARATHON data from (top) isospin-
independent off-shell nucleon modification and (bottom) isospin-dependent off-
shell nucleon modification. Constant-in-x and linear-in-x off-shell function
results are shown with the spectral function and light-cone convolution in
different lines. The 90% C.I. band is taken from the trial with the largest
uncertainty [(top) linear-in-x, spectral function band is used, (bottom)
linear-in-x, light-cone band is used].

To summarize, we have studied the impact of the MARATHON results on
constraints on both the free neutron-to-proton structure function ratio,
$F_{2}^{n}/F_{2}^{p}$, as $x_{B}\rightarrow 1$, and the virtuality dependence
of the nucleon structure modification. Whereas MARATHON chose to use the
single KP model for the virtuality dependence to extract
$F_{2}^{n}/F_{2}^{p}\big{|}_{x_{B}\to 1}$, we have examined the impact of a
wide range of possible forms for this unknown dependence. We find that for an
isospin-independent modification the inclusion of MARATHON data results in a
ratio $F_{2}^{n}/F_{2}^{p}\big{|}_{x_{B}\to 1}=0.43\pm 0.04$, regardless of
the specific virtuality-dependence. However, for an isospin-dependent
modification, the constraints are less stringent
($F_{2}^{n}/F_{2}^{p}\big{|}_{x_{B}\to 1}\approx 0.4\pm 0.1$) and show
variation based on the specific model used. Furthermore, the inclusion of the
MARATHON data does little to constrain the virtuality-dependence of nucleon
modification, and therefore sheds little light on the nature of the EMC
Effect. A wide range of forms and parameters for the nucleon modification
function can easily accommodate the MARATHON, BONuS, and Seely data. We see
this as strong motivation to look to new observables beyond inclusive DIS to
help constrain models of the EMC Effect, such as spectator-tagged DIS, on
which the results of the BONuS12 Bueltmann  _et al._ (2010) and BAND and LAD
Hen  _et al._ (2015); Segarra  _et al._ (2020c) experiments are anticipated.

###### Acknowledgements.

We thank a bunch of people for insightful discussions. This work was supported
by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics
under Award Numbers DE-FG02-94ER40818, DE-SC0020240, DE-FG02-96ER-40960, DE-
FG02-93ER40771.

## References

  * Bashir  _et al._ (2012) A. Bashir, L. Chang, I. Cloet, B. El-Bennich, Y. Liu, C. Roberts, and P. Tandy, Commun. Theor. Phys. 58, 79 (2012), arXiv:1201.3366 [nucl-th] .
  * Roberts  _et al._ (2013) C. D. Roberts, R. J. Holt, and S. M. Schmidt, Phys. Lett. B727, 249 (2013), arXiv:1308.1236 [nucl-th] .
  * Bjorken and Paschos (1969) J. D. Bjorken and E. A. Paschos, Phys. Rev. 185, 1975 (1969).
  * Segarra  _et al._ (2020a) E. P. Segarra, A. Schmidt, T. Kutz, D. W. Higinbotham, E. Piasetzky, M. Strikman, L. B. Weinstein, and O. Hen, Phys. Rev. Lett. 124, 092002 (2020a).
  * Dulat  _et al._ (2016) S. Dulat, T.-J. Hou, J. Gao, M. Guzzi, J. Huston, P. Nadolsky, J. Pumplin, C. Schmidt, D. Stump, and C. P. Yuan, Phys. Rev. D93, 033006 (2016), arXiv:1506.07443 [hep-ph] .
  * Accardi  _et al._ (2016) A. Accardi, L. T. Brady, W. Melnitchouk, J. F. Owens, and N. Sato, Phys. Rev. D93, 114017 (2016), arXiv:1602.03154 [hep-ph] .
  * Arrington  _et al._ (2012) J. Arrington, J. G. Rubin, and W. Melnitchouk, Phys. Rev. Lett. 108, 252001 (2012), arXiv:1110.3362 [hep-ph] .
  * Abrams  _et al._ (2021) D. Abrams _et al._ (MARATHON), (2021), arXiv:2104.05850 [hep-ex] .
  * Griffioen  _et al._ (2015) K. A. Griffioen _et al._ , Phys. Rev. C92, 015211 (2015), arXiv:1506.00871 [hep-ph] .
  * Seely  _et al._ (2009) J. Seely _et al._ , Phys. Rev. Lett. 103, 202301 (2009).
  * Close (1979) F. E. Close,  _An Introduction to Quarks and Partons_ (Academic Press, London, 1979).
  * Close (1973) F. E. Close, Phys. Lett. 43B, 422 (1973).
  * Carlitz (1975) R. D. Carlitz, Phys. Lett. 58B, 345 (1975).
  * Farrar and Jackson (1975) G. R. Farrar and D. R. Jackson, Phys. Rev. Lett. 35, 1416 (1975).
  * Hen  _et al._ (2017) O. Hen, G. A. Miller, E. Piasetzky, and L. B. Weinstein, Rev. Mod. Phys. 89, 045002 (2017).
  * Baillie  _et al._ (2012) N. Baillie _et al._ (CLAS Collaboration), Phys. Rev. Lett. 108, 142001 (2012), [Erratum: Phys. Rev. Lett.108,199902(2012)], arXiv:1110.2770 [nucl-ex] .
  * Tkachenko  _et al._ (2014) S. Tkachenko _et al._ (CLAS Collaboration), Phys. Rev. C 89, 045206 (2014).
  * Bueltmann  _et al._ (2010) S. Bueltmann _et al._ , “The Structure of the Free Neutron at Large x-Bjorken,” Jefferson Lab experiment E12-10-102 (2010).
  * Petratos  _et al._ (2010) G. G. Petratos _et al._ , Jefferson Lab PAC36 Proposal (2010).
  * Kulagin and Petti (tion) S. A. Kulagin and R. Petti, (private communication).
  * Kulagin and Petti (2010) S. A. Kulagin and R. Petti, Phys. Rev. C82, 054614 (2010), arXiv:1004.3062 [hep-ph] .
  * Segarra  _et al._ (2020b) E. P. Segarra, J. R. Pybus, F. Hauenstein, D. W. Higinbotham, G. A. Miller, E. Piasetzky, A. Schmidt, M. Strikman, L. B. Weinstein, and O. Hen, (2020b), arXiv:2006.10249 [hep-ph] .
  * Frankfurt and Strikman (1985) L. L. Frankfurt and M. I. Strikman, Nucl. Phys. B250, 143 (1985).
  * Frankfurt and Strikman (1987) L. Frankfurt and M. Strikman, Phys. Lett. B183, 254 (1987).
  * Frankfurt and Strikman (1988) L. Frankfurt and M. Strikman, Phys. Rep. 160, 235 (1988).
  * Akulinichev  _et al._ (1985) S. Akulinichev, S. Shlomo, S. Kulagin, and G. Vagradov, Phys. Rev. Lett. 55, 2239 (1985).
  * Dunne and Thomas (1985) G. Dunne and A. Thomas, Nucl.Phys. A455, 701 (1985).
  * Kulagin and Petti (2006) S. Kulagin and R. Petti, Nuclear Physics A 765, 126 (2006).
  * Airapetian  _et al._ (2011) A. Airapetian, N. Akopov, and Z. e. a. Akopov, J. High Energ. Phys. 126 (2011), 10.1007/JHEP05(2011)126.
  * Arneodo  _et al._ (1995) M. Arneodo _et al._ (NMC), Physics Letters B 364, 107 (1995).
  * Hen  _et al._ (2015) O. Hen _et al._ , “In Medium Proton Structure Functions, SRC, and the EMC effect,” Jefferson Lab experiment E12-11-003A (2015).
  * Segarra  _et al._ (2020c) E. P. Segarra _et al._ , Nucl. Instrum. Meth. A 978, 164356 (2020c), arXiv:2004.10339 [physics.ins-det] .