# Two results on character codegrees

Yang Liu  and  Yong Yang  School of Mathematical Science, Tianjin Normal
University, Tianjin 300387, P. R. China<EMAIL_ADDRESS>Department of
Mathematics, Texas State University, San Marcos, TX 78666, USA
<EMAIL_ADDRESS>
Abstract. Let $G$ be a finite group and $\mathrm{Irr}(G)$ be the set of
irreducible characters of $G$. The codegree of an irreducible character $\chi$
of the group $G$ is defined as
$\mathrm{cod}(\chi)=|G:\mathrm{ker}(\chi)|/\chi(1)$. In this paper, we study
two topics related to the character codegrees. Let $\sigma^{c}(G)$ be the
maximal integer $m$ such that there is a member in $\mathrm{cod}(G)$ having
$m$ distinct prime divisors, where
$\mathrm{cod}(G)=\\{\mathrm{cod}(\chi)|\chi\in\mathrm{Irr}(G)\\}$. One is
related to the codegree version of the Huppert’s $\rho$-$\sigma$ conjecture
and we obtain the best possible bound for $|\pi(G)|$ under the condition
$\sigma^{c}(G)=2,3,$ and $4$ respectively. The other is related to the prime
graph of character codegrees and we show that the codegree prime graphs of
several classes of groups can be characterized only by graph theoretical
terms.

Keywords. finite group, character codegree, prime graph

MSC2010. 20C15, 20D25

##  1\. Introduction

Throughout this paper, $G$ is a finite group and $\mathrm{Irr}(G)$ is the set
of irreducible characters of $G$. The concept of character codegree (defined
as $|G|/\chi(1)$ for any nonlinear irreducible character $\chi$ of $G$) was
first introduced in [2] to characterize the structure of finite groups.
However for nonlinear character $\chi\in\mathrm{Irr}(G/N)$, where $N$ is a
nontrivial normal subgroup of $G$, $\chi$ will have two different codegrees
when it is considered as the character of $G$ and $G/N$ respectively. So the
concept was improved as $\mathrm{cod}(\chi)=|G:\mathrm{ker}(\chi)|/\chi(1)$
for any character $\chi\in\mathrm{Irr}(G)$ by Qian, Wang, and Wei in [13]. Now
many properties of codegree have been studied, such as the relationship
between the codegrees and the element orders [8, 14], codegrees of $p$-groups
[3, 4], and groups with few codegrees [1, 11, 12].

In this paper, we study two topics related to the character codegrees. Set
${\operatorname{cod}}(G)=\\{\mathrm{cod}(\chi)\ |\ \chi\in\mathrm{Irr}(G)\\}$.
Denote by $\pi(n)$ be the set of prime divisors of rational integer $n$ and we
use $\pi(G)$ instead of $\pi(|G|)$. Let $\sigma^{c}(G)$ be the maximal integer
$m$ such that there is a member in $\rm{cod}$$(G)$ having $m$ distinct prime
divisors. The first topic is about the following question which is raised in
[13, Question A] and called the codegree version of the Hupperts’s
$\rho$-$\sigma$ conjecture: Is there any constant $k$ (independent of $G$)
such that $|\pi(G)|\leq k\sigma^{c}(G)$? In particular, is it true that
$|\pi(G)|\leq 4$ when $\sigma^{c}(G)=2$?

We note that the first question was answered in [15] and further improved in
[16]. In this paper, we show that the second question is not always true in
general and we obtain the best possible bound for $|\pi(G)|$ under the
condition $\sigma^{c}(G)=2,3,$ and $4$ respectively.

######  Theorem 1.1.

Let $G$ be a finite group, then

(i) $|\pi(G)|\leq 5$ when $\sigma^{c}(G)=2$.

(ii) $|\pi(G)|\leq 8$ when $\sigma^{c}(G)=3$.

(iii) $|\pi(G)|\leq 12$ when $\sigma^{c}(G)=4$.

The element order prime graph (Gruenberg-Kegel graph) of a finite group is the
(simple undirected) graph whose vertices are the prime numbers dividing the
order of the group, with two vertices being linked by an edge if and only if
their product divides the order of some element of the group. Gruber et al.
characterized the element order prime graph of solvable groups only using
graph theoretical terms in [6]. Recently Florez et al. gave complete
characterizations for the element order prime graphs of several classes of
groups, such as groups of square-free order, meta-nilpotent groups, groups of
cube-free order, and, for any $n\in N$, solvable groups of $n^{th}$-power-free
order in [5].

The codegree prime graph built on ${\operatorname{cod}}(G)$, that we denote by
$\Delta_{{\operatorname{cod}}}(G)$, is the (simple undirected) graph whose
vertices are the prime divisors of the numbers in ${\operatorname{cod}}(G)$,
and two distinct vertices $p$, $q$ are adjacent if and only if $pq$ divides
some number in ${\operatorname{cod}}(G)$. The second topic of this paper
concerns the relation of solvable groups with the corresponding prime graph of
character codegrees and we obtain several results parallel to the results in
[5, 6].

##  2\. The codegree analogue of the Huppert’s $\rho$-$\sigma$ conjecture

We first introduce some notation. Let $G$ be a group and $N\unlhd G$. For
$\chi\in\mathrm{Irr}(G)$, $\mathrm{Irr}(\chi|_{N})$ is the set of irreducible
constitutes of the restriction of $\chi$ to $N$.

######  Lemma 2.1 (Lemma 1.8 of [9]).

Let $|G|=q_{1}\cdots q_{n}r_{1}\cdots r_{m}$ with mutually distinct primes
$q_{i},r_{j}$ $(n,m\in\mathbb{N})$. Assume that $|F(G)|=r_{1}r_{2}\cdots
r_{m}$ and $p$ is a prime with $|G|\mid p-1$. Then there exists a faithful
irreducible $GF(p)G$-module $V$ such that $F(G)$ acts fixed point freely on
$V$.

######  Example 2.1 (Example 1.9 of [9]).

Let $q_{1},q_{2},q_{3},q_{4},r_{1},r_{2},r_{3},r_{4}$ be mutually distinct
prime numbers where the $q_{i}$ may be arbitrarily chosen and the $r_{i}$ may
have the following properties:

$q_{1}\mid r_{1}-1$, $q_{2}\mid r_{2}-1$, $q_{1}q_{2}q_{3}\mid r_{3}-1$,
$q_{2}q_{4}\mid r_{4}-1$.

The $r_{i}(i=1,2,3,4)$ surely exist because of the Dirichlet’s theorem on
primes in arithmetic progression. We want $G/G^{\prime\prime}$ to have the
following structure:

$G/G^{\prime}\cong Q_{1}\times Q_{2}\times Q_{3}\times Q_{4}$ with $Q_{i}$
cyclic of order $q_{i}$,

$G^{\prime}/G^{\prime\prime}\cong R_{1}\times R_{2}\times R_{3}\times R_{4}$
with $R_{i}$ cyclic of order $r_{i}$, and $Q_{i}\leq N_{G}(R_{j})$ for all
$i,j$. Then $Q_{i}$ can only act trivially or fixed point freely on $R_{j}$
$(i,j\in\\{1,2,3,4\\})$, and we demand:

$Q_{1}$ acts fixed point freely only on $R_{1}$ and $R_{3}$,

$Q_{2}$ acts fixed point freely only on $R_{2}$, $R_{3}$, and $R_{4}$,

$Q_{3}$ acts fixed point freely only on $R_{3}$, and finally

$Q_{4}$ acts fixed point freely only on $R_{4}$.

This is possible because of our number theoretical restriction on the $r_{i}$.
Now let $s_{1},s_{2},s_{3},s_{4}$ be mutually distinct primes such that for
$i=1,2,3,4$, $s_{i}-1$ is divisible by
$q_{1}q_{2}q_{3}q_{4}r_{1}r_{2}r_{3}r_{4}$. (By Dirichlet’s theorem such
$s_{i}$ exist.) Then let $G^{\prime\prime}=V_{1}\times V_{2}\times V_{3}\times
V_{4}$ with irreducible $GF(s_{i})$-modules $V_{i}$, where the $V_{i}$ may be
constructed as follows:

  1. (1)

Let $V_{1}$ be the faithful irreducible $GF(s_{1})Q_{1}R_{1}$-module (existing
by Lemma 2.1), on which $R_{1}$ acts fixed point freely. Let $Q_{2}$, $Q_{3}$,
and $Q_{4}$ act fixed point freely on $V_{1}$ by multiplication with elements
of $GF(s_{1})$ (i.e. $Q_{2}Q_{3}Q_{4}$ is represented by multiples of the
identity matrix), and $R_{2}R_{3}R_{4}$ may act trivially on $V_{1}$. So we
have

$C_{G/G^{\prime\prime}}(V_{1})\cong R_{2}\times R_{3}\times R_{4}$, and

$G/C_{G}(V_{1})\cong Q_{1}R_{1}\times Q_{2}\times Q_{3}\times Q_{4}$.

  2. (2)

As in (1) let $V_{2}$ be the $GF(s_{2})Q_{2}R_{2}$-module, on which $R_{2}$
acts fixed point freely, and we demand

$C_{G/G^{\prime\prime}}(V_{2})\cong R_{1}\times R_{3}\times R_{4}$, and

$G/C_{G}(V_{2})\cong Q_{2}R_{2}\times Q_{1}\times Q_{3}\times Q_{4}$, where
$Q_{1},Q_{3}$ and $Q_{4}$ act fixed point freely on $V_{2}$ by multiplication
with elements of $GF(s_{2})$.

  3. (3)

Let $V_{3}$ be the faithful, irreducible
$GF(s_{3})Q_{1}Q_{2}Q_{3}R_{1}R_{2}R_{3}$-module, on which
$R_{1}R_{2}R_{3}=F(Q_{1}Q_{2}Q_{3}R_{1}R_{2}R_{3})$ acts fixed point freely.
Let $Q_{4}$ act fixed point freely on $V_{3}$ by multiplication with elements
of $GF(s_{3})$, and let $R_{4}$ act trivially on $V_{3}$. Then

$C_{G/G^{\prime\prime}}(V_{3})\cong R_{4}$, and

$G/C_{G}(V_{3})\cong Q_{1}Q_{2}Q_{3}R_{1}R_{2}R_{3}\times Q_{4}$.

  4. (4)

As in (3) let $V_{4}$ be the $GF(s_{4})Q_{1}Q_{2}Q_{4}R_{1}R_{2}R_{4}$-module,
so that $R_{1}R_{2}R_{4}$ acts fixed point freely on it, and we demand

$C_{G/G^{\prime\prime}}(V_{4})\cong R_{3}$, and

$G/C_{G}(V_{3})\cong Q_{1}Q_{2}Q_{4}R_{1}R_{2}R_{4}\times Q_{3}$, where
$Q_{3}$ acts fixed point freely on $V_{4}$ by multiplication with elements of
$GF(s_{4})$.

######  Proposition 2.1.

Let $G$ be the group constructed in Example 2.1. Then $|\sigma^{c}(G)|=4$.

###### Proof.

It can be checked that $F(G)=V_{1}V_{2}V_{3}V_{4}$ and
$F_{2}(G)/F(G)=F(G/F(G))=R_{1}R_{2}R_{3}R_{4}F(G)/F(G)$. Next we consider
prime divisors of $\mathrm{cod}(\chi)$ case by case for any
$\chi\in\mathrm{Irr}(G)$.

Case (a). $G^{\prime\prime}\not\leq\mathrm{ker}(\chi)$.

Choose $\lambda\in\mathrm{Irr}(\chi|_{F(G)})$. We may assume
$\lambda=\lambda_{1}\cdot\lambda_{2}\cdot\lambda_{3}\cdot\lambda_{4}$, where
$\lambda_{i}\in\mathrm{Irr}(V_{i})$.

Subcase (a1). Suppose that $\lambda_{i}\neq 1$, $i=1,2,3,4$.

It can be checked that $I_{G}(\lambda)=F(G)$ and $\chi(1)=|G/F(G)|$. Thus
$\pi(\mathrm{cod}(\chi))=\\{s_{1},s_{2},s_{3},s_{4}\\}$.

Subcase (a2). Suppose that one of the four characters $\lambda_{i}$ is trivial
and the other three characters are nontrivial.

If $\lambda_{4}=1$, it can be checked that $I_{G}(\lambda)=F(G)R_{4}$ and
$\chi(1)=|G/(F(G)R_{4})|$. Since $V_{4}\leq\mathrm{ker}(\lambda)$, we have
$V_{4}\leq\mathrm{ker}(\chi)$ and
$\pi(\mathrm{cod}(\chi))\subseteq\\{s_{1},s_{2},s_{3},r_{4}\\}$. Similarly, we
can check for subcases of $\lambda_{i}=1$, $i=1,2,3$ and also obtain that
$|\pi(\mathrm{cod}(\chi))|\leq 4$.

Subcase (a3). Suppose that two of the four characters $\lambda_{i}$ are
trivial and the other two characters are nontrivial.

If $\lambda_{3}$ and $\lambda_{4}$ are trivial characters, it can be checked
that $I_{G}(\lambda)\leq F(G)(R_{3}R_{4})$ and
$|G|/\chi(1)\mid|F(G)R_{3}R_{4}|$. Since
$V_{3}V_{4}\leq\mathrm{ker}(\lambda)$, we have
$V_{3}V_{4}\leq\mathrm{ker}(\chi)$ and
$\pi(\mathrm{cod}(\chi))\subseteq\\{s_{1},s_{2},r_{3},r_{4}\\}$. Similarly, we
can check for other subcases and also obtain that
$|\pi(\mathrm{cod}(\chi))|\leq 4$.

Subcase (a4). Suppose that three of the four characters $\lambda_{i}$ are
trivial and the remained character is nontrivial.

If $\lambda_{4}\neq 1$, then $T=I_{G}(\lambda)\leq F(G)R_{3}(Q_{1}Q_{2}Q_{4})$
and $\lambda$ extends to $\theta\in\mathrm{Irr}(T)$. By Clifford’s theorem,
there exists $\beta\in\mathrm{Irr}(T/F(G))$ such that
$\chi=(\theta\cdot\beta)^{G}$. Since $V_{1}V_{2}V_{3}\leq\mathrm{ker}(\chi)$,

$\mathrm{cod}(\chi)=\frac{|G|}{\mathrm{ker}(\chi)\chi(1)}=\frac{|G|}{|\mathrm{ker}(\chi)||G:T|\beta(1)}=\frac{|T|}{|\mathrm{ker}(\chi)|\beta(1)}\mid|V_{4}R_{3}(Q_{1}Q_{2}Q_{4})|.$

If $T\not\leq F(G)R_{3}(Q_{1}Q_{2}Q_{4})$ or $\beta(1)>1$, then
$|\pi(\mathrm{cod}(\chi))|\leq 4$. Next we assume
$T=F(G)R_{3}(Q_{1}Q_{2}Q_{4})$ and $\beta(1)=1$. Since
$R_{3}=[R_{3},Q_{1}]\leq T^{\prime}\leq\mathrm{ker}(\theta\cdot\beta)$ and
$V_{1}V_{2}V_{3}R_{3}$ is a normal subgroup of $G$, we have
$V_{1}V_{2}V_{3}R_{3}\leq\mathrm{ker}(\theta\cdot\beta)$ and
$V_{1}V_{2}V_{3}R_{3}\leq\mathrm{ker}(\chi)$. Therefore
$\pi(\mathrm{cod}(\chi))\subseteq\\{s_{4},q_{1},q_{2},q_{4}\\}$.

Case (b). $G^{\prime\prime}\leq\mathrm{ker}(\chi)$.

Choose $\lambda\in\mathrm{Irr}(\chi|_{F_{2}(G)})$. We may assume
$\lambda=\lambda_{1}\cdot\lambda_{2}\cdot\lambda_{3}\cdot\lambda_{4}$, where
$\lambda_{i}\in\mathrm{Irr}(R_{i})$. All cases can be handled with the same
method as before and we take the following case as an example. Suppose that
$\lambda_{1},\lambda_{2}$ are nontrivial characters and
$\lambda_{3},\lambda_{4}$ are trivial, then
$I_{G}(\lambda)=F_{2}(G)Q_{3}Q_{4}$ and $\chi(1)=q_{1}q_{2}$. Since
$F(G)R_{3}R_{4}\leq\mathrm{ker}(\chi)$, we have
$\pi(\mathrm{cod}(\chi))\subseteq\\{r_{1},r_{2},q_{3},q_{4}\\}$. ∎

By the same method, we can obtain the following result.

######  Proposition 2.2.

Let $H=Q_{1}Q_{2}R_{1}R_{2}V_{1}$ and
$K=Q_{2}Q_{3}Q_{4}R_{2}R_{4}V_{1}V_{3}V_{4}$, where $Q_{i},R_{j},V_{k}$ are
from Example 2.1, then $|\sigma^{c}(H)|=2$ and $|\sigma^{c}(K)|=3.$

By the previous examples and the main results of [9, 10], as well as the main
result of [8], Theorem 1.1 can be easily checked. Note that all those upper
bounds are the best possible.

##  3\. The codegree prime graphs of solvable groups

The following three results are about the prime graphs (element order version)
of solvable groups.

######  Proposition 3.1 (Theorem 2 of [6]).

An unlabeled graph $F$ is isomorphic to the prime graph of some solvable group
if and only if its complement $\overline{F}$ is 3-colorable and triangle-free.

######  Proposition 3.2 (Theorem 3.1 of [5]).

Let $\mathcal{C}$ be any class of groups such that

{$G$: $G$ has square-free order}$\subseteq\mathcal{C}\subseteq$ {$G$: $G$
metanilpotent}.

Then $F$ is isomorphic to the prime graph of some $G\in\mathcal{C}$ if and
only if $\overline{F}$ is bipartite.

######  Proposition 3.3 (Theorem 4.3 of [5]).

$F$ is isomorphic to the prime graph of a solvable group of $n^{th}$-power-
free order if and only if $\overline{F}$ satisfies the following conditions.

(1) Triangle-free.

(2) There exists a $3$-coloring of $\overline{F}$ by Red, Green, and Blue and
a way to label each red vertex by a distinct prime number such that for any
$\pi$ a subset of those primes whose product is at least $n$, we have that in
the canonical orientation, no blue vertex is simultaneously the end of
directed $2$-paths starting from each of the red vertices in $\pi$.

In [14], Qian proved that if $G$ is a solvable group and $g\in G$, then there
is an irreducible character $\chi$ of $G$ such that $p$ divides
$\mathrm{cod}(\chi)$ for any prime divisor $p$ of the order of $g$. A while
later, Isaacs proved that the result is indeed true for any finite group in
[8]. In view of this result, we see that the element order prime graph of a
finite group $G$ is a subgraph of its codegree prime graph. Next we show the
element order prime graph and codegree prime graph coincide for various types
of solvable groups.

######  Proposition 3.4.

Let $G$ be a solvable group such that every minimal normal subgroup is a Sylow
subgroup and $|G/F(G)|$ is square-free. For any prime pairs $(p,q)$ if
$pq\mid\mathrm{cod}(\chi)$ for some $\chi\in\mathrm{Irr}(G)$, then there
exists an element $g\in G$ such that $pq\mid|g|$.

###### Proof.

We proceed by the induction on the order of $G$. If $\mathrm{ker}(\chi)$ is
nontrivial, then it can be checked that every minimal normal subgroup of
$G/\mathrm{ker}(\chi)$ is a Sylow subgroup and
$(G/\mathrm{ker}(\chi))/F(G/\mathrm{ker}(\chi))$ is square-free. Then there
exists an element $g\mathrm{ker}(\chi)\in G/\mathrm{ker}(\chi)$ such that
$pq\mid|g\mathrm{ker}(\chi)|$ by induction. Therefore $pq\mid|g|$. Next we
assume that $\mathrm{ker}(\chi)=1$.

Let $F(G)=P_{1}\times P_{2}\times\cdots\times P_{t}$, where $P_{i}$ is a Sylow
$p_{i}$-subgroup. Choose $\lambda\in\mathrm{Irr}(\chi|_{F(G)})$ and set
$T=I_{G}(\lambda)$. Then $\chi(1)=|G:T|\theta(1)$, where
$\theta\in\mathrm{Irr}(T|\lambda)$ by Clifford theory and thus
$\mathrm{cod}(\chi)=|T|/\theta(1)$. If $pq\mid|F(G)|$, then there exists an
element of order $pq$. If $p\mid|F(G)|$ and $(q,|F(G)|)=1$, then the order of
$\lambda$ is divisible by $p$ and $q\mid|T|$. Since the actions of $G/F(G)$ on
$\mathrm{Irr}(F(G))$ and $F(G)$ are permutation isomorphic (by Theorem 13.24
of [7]), there exists $h\in F(G)$ such that $p\mid|h|$ and $q\mid|C_{G}(h)|$.
Hence there exists an element $g\in G$ such that $pq\mid|g|$. If $q\mid|F(G)|$
and $(p,|F(G)|)=1$, the proposition is true similarly. Now we can assume
$pq\mid|G/F(G)|$. Let $F_{2}(G)/F(G)=F(G/F(G))$. Since $G/F(G)$ is square-
free, $F_{2}(G)/F(G)$ and $G/F_{2}(G)$ are cyclic groups. If
$pq\mid|F_{2}(G)/F(G)|$ or $pq\mid|G/F_{2}(G)|$, the proposition is true.
Without loss of generality we assume that $p\mid|F_{2}(G)/F(G)|$ and
$q\mid|G/F_{2}(G)|$. Choose a Sylow $p$-subgroup $P\leq T$ and set $N=F(G)P$.
Let $\alpha\in\mathrm{Irr}(\chi|_{N})$ such that
$\lambda\in\mathrm{Irr}(\alpha|_{F(G)})$, then $\alpha$ is an extension of
$\lambda$. Since $N^{\prime}\leq\mathrm{ker}(\alpha)$ and $N\unlhd G$,
$N^{\prime}\leq\mathrm{ker}(\chi)$ and $N^{\prime}=1$. Hence $P\leq
C_{G}(F(G))=F(G)$, a contradiction with $(p,|F(G)|)=1$. ∎

Inspired by the previous results, we obtain similar results on the prime graph
of codegrees.

######  Proposition 3.5.

An unlabeled graph $F$ is isomorphic to the codegree prime graph of some
solvable group if and only if its complement $\overline{F}$ is 3-colorable and
triangle-free.

###### Proof.

First, we assume $F$ is isomorphic to the codegree prime graph of a solvable
group $G$. Let $F_{1}$ be the prime graph of $G$. Then $\overline{F_{1}}$ is
3-colorable and triangle-free by Proposition 3.1. Since $F_{1}$ is subgraph of
$F$, $\overline{F}$ is a subgraph of $\overline{F_{1}}$. Hence $\overline{F}$
is 3-colorable and triangle-free.

Conversely, if $\overline{F}$ is 3-colorable and triangle-free, then a
solvable group $G$ can be constructed such that $F$ is isomorphic to the prime
graph of $G$ (The details on the structure of $G$ can be found in Theorem 2.8
of [6]). It can be checked that every minimal normal subgroup of $G$ is a
Sylow subgroup and $|G/F(G)|$ is square-free. Then
$\Delta_{{\operatorname{cod}}}(G)$ is the same as its prime graph by
Proposition 3.4. Therefore $F$ is isomorphic to
$\Delta_{{\operatorname{cod}}}(G)$. ∎

######  Proposition 3.6.

Let $\mathcal{C}$ be any class of groups such that

{$G$: $G$ has square-free order}$\subseteq\mathcal{C}\subseteq$ {$G$: $G$
metanilpotent}.

Then $F$ is isomorphic to the codegree prime graph of some $G\in\mathcal{C}$
if and only if $\overline{F}$ is bipartite.

###### Proof.

First, we assume $F$ is isomorphic to the codegree prime graph of a solvable
group $G$. Let $F_{1}$ be the prime graph of $G$. Then $\overline{F_{1}}$ is
bipartite by Proposition 3.2. Since $F_{1}$ is subgraph of $F$, $\overline{F}$
is a subgraph of $\overline{F_{1}}$. Hence $\overline{F}$ is bipartite.

Conversely, if $\overline{F}$ is 3-colorable and triangle-free, then a
solvable group $G$ of square-free order can be constructed such that $F$ is
isomorphic to the prime graph of $G$ (The details on the structure of $G$ can
be found in Theorem 3.1 of [5]). Since $G$ has square-free order,
$\Delta_{{\operatorname{cod}}}(G)$ is the same as its prime graph. Therefore
$F$ is isomorphic to $\Delta_{{\operatorname{cod}}}(G)$. ∎

######  Proposition 3.7.

$F$ is isomorphic to the codegree prime graph of a solvable group of
$n^{th}$-power-free order if and only if $\overline{F}$ satisfies the
following conditions.

(1) Triangle-free;

(2) There exists a $3$-coloring of $\overline{F}$ by Red, Green, and Blue and
a way to label each red vertex by a distinct prime number such that for any
$\pi$ a subset of those primes whose product is at least $n$, we have that in
the canonical orientation, no blue vertex is simultaneously the end of
directed $2$-paths starting from each of the red vertices in $\pi$.

###### Proof.

First, we assume $F$ is isomorphic to the codegree prime graph of solvable
group $G$ with $n^{th}$-power-free order. Let $F_{1}$ be the prime graph of
$G$. Then $\overline{F_{1}}$ satisfies the conditions in Proposition 3.3, so
does $\overline{F}$ for $\overline{F}$ is a subgraph of $\overline{F_{1}}$.

Conversely, if $\overline{F}$ satisfies the conditions, then by Theorem 4.3 of
[5] a solvable group $G$ can be constructed such that $F$ is isomorphic to the
prime graph of $G$ which is still from Theorem 2.8 of [6]. Then
$\Delta_{{\operatorname{cod}}}(G)$ is the same as its prime graph by
Proposition 3.4. Therefore $F$ is isomorphic to
$\Delta_{{\operatorname{cod}}}(G)$. ∎

Similar with Corollaries 4.4-4.7 in [5], we can obtain the following results.

######  Corollary 3.1.

$F$ is isomorphic to the codegree prime graph of a solvable group of
$n^{th}$-power-free order if and only if $\overline{F}$ satisfies the
following conditions.

(1) Triangle-free;

(2) There exists a 3-coloring of $\overline{F}$ by Red, Green, and Blue such
that we can label the red vertices by the first m primes such that

(a) The primes are less than $n$;

(b) In the canonical orientation, no blue vertex is simultaneously the end of
directed $2$-paths starting from each of the red vertices in $\pi$, where
$\pi$ is any subset of the first $m$ primes whose product is at least $n$.

######  Corollary 3.2.

$F$ is isomorphic to the codegree prime graph of a solvable group of square-
free order if and only if $\overline{F}$ is $2$-colorable.

######  Corollary 3.3.

$F$ is isomorphic to the codegree prime graph of a solvable group of cube free
order if and only if $\overline{F}$ is triangle free and is $2$-colorable
after removing one vertex.

######  Corollary 3.4.

The following are equivalent.

(1) $F$ is isomorphic to the codegree prime graph of a solvable group of
fourth-power-free order.

(2) $F$ is isomorphic to the codegree prime graph of a solvable group of
fifth-power-free order.

(3) $\overline{F}$ is $3$-colorable, triangle free and satisfies one of the
following conditions.

(a) $\overline{F}$ is $2$-colorable after removing one vertex;

(b) There exists a $3$-coloring of $\overline{F}$ by Red, Green, and Blue such
that there are exactly $2$ red vertices. Furthermore, in the canonical
orientation, no vertex is the end of a directed $2$-path starting from both of
the red vertices.

Acknowledgments

The project is supported by NSFC (Grant Nos. 11671063, 11701421, and
11871011), the Simons Foundation (No. 499532), and the Science & Technology
Development Fund of Tianjin Education Commission for Higher Education
(2020KJ010).

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