# Graphs with Large Girth and Small Cop Number

Alexander Clow
Department of Mathematics
Simon Fraser University
Burnaby, British Columbia, Canada
<EMAIL_ADDRESS>

###### Abstract.

In this paper we consider the cop number of graphs with no, or few, short
cycles. We show that when the girth of $G$ is at least $7$ and the minimum
degree is sufficiently large, $\delta\geq(\ln{n})^{\frac{1}{1-\alpha}}$ where
$\alpha\in(0,1)$, then $c(G)=o(n\delta^{1-\lfloor\frac{g}{4}\rfloor})$ as
$n\rightarrow\infty$. This extends work of Frankl and implies that if $G$ is
large and dense in the sense that $\delta\geq n^{\frac{2+o(1)}{g-1}}$ while
also having girth $g\geq 7$, then $G$ satisfies Meyniel’s conjecture, that is
$c(G)=O(\sqrt{n})$. Moreover, it implies that if $G$ is large and dense in the
sense that there $\delta\geq n^{\varepsilon}$ for some $\varepsilon>0$, while
also having girth $g\geq 7$, then there exists an $\alpha>0$ such that
$c(G)=O(n^{1-\alpha})$, thereby satisfying the weak Meyniel’s conjecture. Of
course, this implies similar results for dense graphs with small, that is
$O(n^{1-\alpha})$, numbers of short cycles, as each cycle can be broken by
adding a single cop. We also, show that there are graphs $G$ with girth $g$
and minimum degree $\delta$ such that the cop number is at most
$o(g(\delta-1)^{(1+o(1))\frac{g}{4}})$. This resolves a recent conjecture by
Bradshaw, Hosseini, Mohar, and Stacho, by showing that the constant
$\frac{1}{4}$ cannot be improved in the exponent of a lower bound
$c(G)\geq\frac{1}{g}(\delta-1)^{\lfloor\frac{g-1}{4}\rfloor}$.

##  1\. Introduction

In this paper we consider the game of _cops and robbers_ on graphs. Cops and
robbers is a 2-player game played on a graph $G=(V,E)$. To being the game, the
cop player places $k$ cops onto vertices of the graph, then the robber player
chooses a vertex of the graph to place the robber at the beginning of the
game. After this players take turns moving their pieces from the current
vertex they occupy, say $v$, to a vertex in $N[v]$. Note this allows a player
to not move a given cop or robber, should they desire to do so. The cops win
if after finitely many moves a cop is able to move onto the vertex currently
occupied by the robber, which we call capturing the robber, while the robber
wins if there exists a strategy which indefinitely evades capture. The least
number of cops required to win the game no matter which vertex the robber
begins on is known as the cop number of a graph, denoted $c(G)$. Suppose all
graphs we consider are connected.

Cops and robbers was introduced independently by Quilliot [26], and Nowakowski
and Winkler [23]; while the cop number of a graph, was first introduced by
Aigner and Fromme [1]. In recent decades cops and robbers has seen a
remarkable amount of attention with many general results as well as results
for specific graph classes appearing in the literature. For instance, the cop
number of planar graphs [1], graphs of higher genus [6, 8, 27, 28], Cayley
graphs [9, 15], graph products [22], and random graphs [4, 21, 25] have all
been extensively studied. Significant attention has also been paid to
computational questions involving cop number, with MacGillivray and Clarke
[13] showing that deciding if a graph is $k$-cop win is fixed parameter
polynomial time in the order of the graph and $k$, while Kinnersley [17]
proved that determining if $c(G)\leq k$ is EXPTIME-complete when $k$ is not
fixed.

The most famous problem in cops and robbers is undoubtedly _Meyniel’s
conjecture_ , which states that for all graphs $G$, $c(G)=O(\sqrt{n})$. If
correct, then this bound could not be improved as there are known graph
families with $c(G)=\Omega(\sqrt{n})$ [5, 24]. Despite Meyniel’s conjecture
being resolved for a number of graph classes such as Abelian Cayley graphs [9]
and random graphs [4, 25], the larger conjecture remains widely open despite
significant effort [11, 16, 19, 29]. In fact, it remains to be shown that
there exists a $\alpha>0$ such that $c(G)=O(n^{1-\alpha})$, with this problem
sometimes begin dubbed the _weak Meyniel’s conjecture_. For more on Meyniel’s
conjecture see [3].

In this paper we focus on graphs with no, or few short cycles. The _girth_ of
a graph $G$ is the length of the shortest cycle in $G$. Considering graphs of
large girth allows our analysis to avoid complicated local structures
appearing in may graphs. Cops and robbers on graphs with large girth was first
considered by Frankel in [15] who showed that that
$c(G)>(\delta-1)^{(1-o(1))\frac{g}{8}}$ and if $G$ is a $d$-regular graph of
girth $g$, then $c(G)\leq
n(1+(\frac{g}{4}+o(1))\ln{d})d^{-(1-o(1))\frac{g}{4}}$. This lower bound was
recently improved by Bradshaw, Hosseini, Mohar, and Stacho in [10] who showed
that $c(G)\geq\frac{1}{g}(\delta-1)^{\lfloor\frac{g-1}{4}\rfloor}$ and
conjecture that the exponential coefficient of $\frac{1}{4}$ cannot be
improved. This conjecture is quite reasonable as it is implied by combining
Meyniel’s conjecture and a conjecture that exists in folklore stating that
there exist $d$-regular graphs of girth $g$ and order
$d^{(1+o(1))\frac{g}{2}}$. We prove in Theorem 1.4 that this conjecture from
[10] is true. Additionally, we improve Frankl’s upper bound for the cop number
of a graph in terms of girth when the minimum degree of a graph is
sufficiently large as a function of $n$ in Theorem 1.1.

######  Theorem 1.1.

If $G=(V,E)$ be a graph of order $n$ which has girth $g\geq 7$ and
$\delta=\delta(G)\geq(\ln{n})^{\frac{1}{1-\alpha}}$ such that
$\alpha\in(0,1)$, then for all $\varepsilon>0$,

$c(G)\leq(\varepsilon+o(1))n\delta^{1-\lfloor\frac{g}{4}\rfloor}.$

######  Corollary 1.2.

If $G=(V,E)$ has girth $g\geq 7$ and $\delta=\delta(G)\geq n^{x}$ where
$x=\frac{1-y}{\lfloor\frac{g}{4}\rfloor-1}$ is a constant, then
$c(G)=o(n^{y})$.

Thus, we have shown that all large graphs with girth $g\geq 7$ and $\delta\geq
n^{\frac{1}{2(\lfloor\frac{g}{4}\rfloor-1)}}$ must have $c=o(\sqrt{n})$.
Therefore, all such graphs trivially satisfy Meyniel’s conjecture.
Furthermore, Corollary 1.2 implies that if $G$ is a sufficiently large graph
with girth at least $7$ and $\delta\geq n^{\varepsilon}$ where
$\varepsilon>0$, then there exists an $\alpha>0$ such that
$c(G)=o(n^{1-\alpha})$ thereby showing that all such graphs satisfy the weak
Meyniel’s conjecture.

Furthermore, we note that this implies that if $G$ is a graph with at most
$O(n^{1-\alpha})$ cycles of length $g-1$ or less, then we may conclude that if
$\delta\geq n^{\frac{\alpha}{\lfloor\frac{g}{4}\rfloor-o(1)}}$, then
$c(G)=O(n^{1-\alpha})$. This is because choosing one vertex per cycle to form
a set of vertices $S$, then placing a cop at each vertex in $S$, means that

$c(G)\leq|S|+c(G-S)=O(n^{1-\alpha})$

given the cops on $S$ can remain on $S$ for the full game, thereby confining
the robber to $G-S$ where $c(G-S)$ cops will be able to capture the robber.

A $d$-regular graph with girth $g$ of minimal order is called a $(d,g)$-cage
and the order of a $(d,g)$-cage is denoted $n(d,g)$. The problem of
constructing $(d,g)$-cages and determining $n(d,g)$ has been extensively
studied by a number of authors. For a survey on such problems see [14]. Of
particular interest for us is the following bound by Lazebnik, Ustimenko, and
Woldar [18]; let $d\geq 2$ and $g\geq 5$ be integers, and let $q$ denote an
odd prime power for which $d\leq q$, then

$n(d,g)\leq 2dq^{\frac{3}{4}g-a}$

where $a=4,11/\\!4,7/\\!2,13/\\!4$, for $g\equiv 0,1,2,3\mod{4}$ respectively.

######  Corollary 1.3.

If $G=(V,E)$ is a $(p^{k},4t+3)$-cage where $t\geq 1$ is fixed and $p$ is an
odd prime, then $c(G)=o(p^{2tk})=o(\delta^{(1-o(1))\frac{g}{2}})$ as
$k\rightarrow\infty$.

######  Theorem 1.4.

There exists graphs $G$ with $\delta(G)=\delta$ and girth $g$ where

$c(G)=o(g(\delta-1)^{(1+o(1))\frac{g}{4}}).$

##  2\. Proof of Theorem 1.1

As the title suggests in this section we will prove Theorem 1.1. Our proof is
in part probabilistic and proceeds similarly to arguments that appear in [19,
29, 10]. For readers unfamiliar with the probabilistic method we recommend [2]
as a reference. Additionally, we adopt notation specific to cops and robbers
from [7]. Observe that for a graph $G=(V,E)$ and a vertex $v\in V$ we define
$B_{r}(v):=\\{u\in V:\operatorname{dist}(u,v)\leq r\\}$.

######  Lemma 2.1 ([1]).

Let $G=(V,E)$ be graph. If $p=v_{0},\dots,v_{k}$ is a shortest path from
$v_{0}$ to $v_{k}$, then $p$ is $1$-guardable.

Note that Lemma 2.1 is a standard tool used to upper bound cop number. For
instance Lemma 2.1 is used by Aigner and Fromme in [1] to show that every
planar graphs has cop number at most $3$, which is tight, while also being
employed in recent research, such as that of Scott and Sudakov [29] who showed
$c(G)\leq n2^{-(1+o(1))\sqrt{\log{n}}}$. The following lemma is less standard
but is easily verified. Surprisingly, these two lemmas are sufficient
prerequisites to prove Theorem 1.1.

######  Lemma 2.2.

Let $G=(V,E)$ be a graph with girth $g=4t+3+r$ where $0\leq r\leq 3$ and
$t\geq 1$. Then, for all $u,v\in V$, if $\operatorname{dist}(v,u)=t$, then
letting $U=B_{t+1}(u)\setminus(\cup_{w\neq u\in V;\\\
\operatorname{dist}(v,w)=t}B_{t+1}(w))$,

$|U|\geq(\delta-1)^{t+1}.$

###### Proof.

Let $G=(V,E)$ be a graph with girth $g=4t+3+r$ where $0\leq r\leq 3$ and
$t\geq 1$. Then, for all $v\in V$, $T_{v}=G[B_{2t+1}(v)]$ is a tree, or the
only cycles in $T_{v}$ contain $v$ and an edge joining two vertices with
distance $2t+1$ from $v$. Let $u$ be a fixed but arbitrary vertex of distance
$t$ from $v$ and let $U=B_{t+1}(u)\setminus(\cup_{w\neq u\in V;\\\
\operatorname{dist}(v,w)=t}B_{t+1}(w))$.

It is clear that there are at least $(\delta-1)^{t+1}$ vertices $z$ in $T_{v}$
with $\operatorname{dist}(z,v)=2t+1$ with a shortest path $v$ in $T_{v}$
containing $u$, hence such leaves are within distance $t+1$ of $u$.
Furthermore such a shortest path connecting $z$ and $v$ must be unique as
girth of $G$ is $g\geq 4t+3$. Then it is clear that for all $w\neq u$ such
that $\operatorname{dist}(w,v)=t$, $w$ has no path to any such vertex $z$ of
length less than or equal to $t+1$ as this would contradict the uniqueness of
a shortest path connecting $z$ and $v$. The result follows immediately. ∎

###### Proof of Theorem 1.1.

Let $G=(V,E)$ be a graph of order $n$ which has girth $g\geq 7$ such that
$\delta=\delta(G)\geq(\ln{n})^{\frac{1}{1-\alpha}}$ for $0<\alpha<1$. Let $C$
be a random subset of $V$ such that for all $v\in V$, $\mathbb{P}(v\in C)=p$
where $p\in(0,1)$ is to be chosen later.

Let $g=4t+3+r$ where $0\leq r\leq 3$. Suppose $v\in V$ be fixed but arbitrary
and let $E_{v}$ be the event that there exits a vertex $u\in V$ where
$\operatorname{dist}(u,v)=t$ and there is no cop within distance $2t+1$ of $v$
whose shortest path to $v$ contains $u$. Then, applying Lemma 2.2, the union
bound, and the fact that $G[B_{2t+1}(v)]$ is a tree, or a graph whose only
cycles contain $v$ and an edge joining two vertices with distance $2t+1$ from
$v$, we can see that,

$\displaystyle\mathbb{P}(E_{v})\leq\Delta(\Delta-1)^{t-1}(1-p)^{(\delta-1)^{t+1}}\leq\Delta(\Delta-1)^{t-1}\operatorname{exp}(-p(\delta-1)^{t+1}).$

Note that $\delta\geq(\ln{n})^{\frac{1}{1-\alpha}}$ implies $n\leq
e^{\delta^{1-\alpha}}$. Hence, letting $p=\varepsilon\delta^{-t}$ where
$\varepsilon>0$ and $\beta>1-\alpha$, and applying the union bound,

$\displaystyle\mathbb{P}(\bigcup_{v\in V}E_{v})\leq n\mathbb{P}(E_{v})$
$\displaystyle\leq
n\Delta(\Delta-1)^{t-1}\operatorname{exp}(-p(\delta-1)^{t+1})$
$\displaystyle\leq n^{t+1}\operatorname{exp}(-p(\delta-1)^{t+1})$
$\displaystyle\leq
n^{t+1}\operatorname{exp}(-(\varepsilon-o(1))\delta^{t+1-t})$
$\displaystyle\leq n^{t+1}\operatorname{exp}(-(\varepsilon-o(1))\delta)$
$\displaystyle\leq\operatorname{exp}((t+1)\delta^{1-\alpha}-(\varepsilon-o(1))\delta)\rightarrow
0$

as $\delta\rightarrow\infty$ with $n\rightarrow\infty$, given our assumption
that $1-\alpha<1$. So asymptotically almost surly for every vertex $v$ and
$u\in V$ where $\operatorname{dist}(u,v)=t$, there is a cop with distance at
most $2t+1$ from $v$ whose shortest path to $v$ contains $u$.

Suppose at the beginning of the game we place $1$ cop on each vertex of $C$
and suppose without loss of generality that the robber begins the game on
vertex $v$. Then asymptotically almost surly the cops will win in at most
$2t+1$ turns by following a strategy where each cop takes a geodesic path
towards $v$. This is because given $G[B_{t}(v)]$ is a tree by our assumption
that $G$ has girth $g\geq 4t+3$ and with high probability the robber will not
be able to escape $B_{t}(v)$ before there is a cop on the unique geodesic path
connecting each $u\in B_{t}(v)$ to $v$; where this cop will either be
occupying the same vertex as the robber or a vertex nearer to $u$ than the the
robber. From here the result follows by Lemma 2.1.

Now we will proceed to bound $|C|$ using Chernoff’s Bound. Note that $|C|$ is
a binomial random variable, hence $\mathbb{E}(|C|)=np=\varepsilon
n\delta^{-t}$. Now applying Chernoff’s Bound,

$\displaystyle\mathbb{P}(|C|\geq\mathbb{E}(|C|)+k)\leq\operatorname{exp}(-\frac{k^{2}}{2(\mathbb{E}(|C|)+\frac{k}{3})})$

so letting $k=\mathbb{E}(|C|)^{c}$ where $\frac{1}{2}<c<1$ it is clear that

$\displaystyle\mathbb{P}(|C|\geq\mathbb{E}(|C|)+k)\rightarrow 0$

as $n\rightarrow\infty$. So asymptotically almost surly,
$|C|\leq\mathbb{E}(|C|)+\mathbb{E}(|C|)^{c}=(\varepsilon+o(1))n\delta^{-t}$.
Noting that $t\geq\lfloor\frac{g}{4}\rfloor-1$ this completes the proof. ∎

Note that our proof also implies that the capture time for
$(\varepsilon+o(1))n\delta^{1-\lfloor\frac{g}{4}\rfloor}$ cops is
approximately $\frac{g}{2}$. This is somewhat surprising, at least to the
author, as this implies the capture time does not necessarily depend on $n$
for such graphs. Due to this we believe that Theorem 1.1 is unlikely to be a
tight. However, it is notable that discovering an improvement with a large
capture time may be difficult as it will likely involve a more complicated
strategy by the cops.

##  3\. Proof of Theorem 1.4

In this section we will show that bipartite Ramanujan graphs introduced by
Lubotsky, Phillips, and Sarnak in [20] have cop number at most
$(\varepsilon+o(1))g\delta^{(1+o(1))\frac{g}{4}}$ for every $\varepsilon>0$.
Note that these are the same graphs considered in [10] to show that there are
graphs who’s cop number is at most $(\delta-1)^{(1+o(1))\frac{3g}{8}}$. Our
argument employs similar ideas to that of [10], hence we will use two lemmas
lemmas which appear in [10]. We begin by defining the _isoperimetric function_
, $\Phi(G,k)$,

$\Phi(G,k):=\min_{S\subset V(G);|S|\leq k}\frac{|N(S)\setminus S|}{|S|}.$

######  Lemma 3.1 ([10]).

Let $G=(V,E)$ be a graph of order $n$. If $\Phi(G,n^{1-\alpha})\geq\beta>0$,
then for all integers $r$, and any nonempty set $S\subset V$,

$|B_{r}(S)|\geq\min\\{n^{1-\alpha},|S|(1+\beta)^{r}\\}.$

Hence, for graphs whose isoperimetric function is large we can lower bound the
size of the $r$-neighbourhood of any small set of vertices $S$. This will be
critical to ensuring that there are sufficiently many cops in a position to
help capturing the robber in the proof of Theorem 1.4.

The next step is to ensure that the graphs we consider have sufficiently large
$\Phi(G,n^{1-\alpha})$. Again we borrow a Lemma from [10] to do this. As this
lemma relies on ideas from spectral graph theory which we do not introduce
here, we refer readers who are curious about or unfamiliarly with spectral
graph theory to [12]. Note that the proof of Lemma 3.2 in [10] relies on a
result of Tanner [30].

######  Lemma 3.2 ([10]).

Let $0<\alpha<1$ be fixed. Let $G$ be a connected and $d$-regular bipartite
graph. Suppose that $G$ has eigenvalues
$d=\lambda_{1}(G)\geq\lambda_{2}(G)\geq\dots\geq\lambda_{n}(G)=-d$. Then

$\Phi(G,n^{1-\alpha})\geq(\frac{d}{\lambda_{2}(G)})^{2}-1-o(1)$

as $n\rightarrow\infty$.

With both of these results in hand we are ready to prove Theorem 1.4.

###### Proof of Theorem 1.4.

Let $G=(V,E)$ be the Ramanujan graph $X^{p,q}$ discovered in [20], where
$d=p+1$, and $p,q\equiv 1\mod{4}$ are prime, and $q\geq\sqrt{p}$ such that the
Legendre symbol of $q$ and $p$ satisfies $(\frac{q}{p})=-1$. Then $G$ is a
$d$-regular Ramanujan graph on $n=q(q^{2}-1)$ vertices with girth
$g\geq\frac{4\log{q}}{\log{p}}-1$ [20]. From the definition of Ramanujan
graphs the second eigenvalue of $G$, $\lambda_{2}$, has $\lambda_{2}^{2}\leq
4(d-1)$.

Applying Lemma 3.2, this implies that as $n\rightarrow\infty$, that is
$q\rightarrow\infty$,

$\Phi(G,n^{1-\alpha})\geq\frac{d^{2}}{4(d-1)}-1-o(1)\geq(1-o(1))\frac{d}{4}$

for any $0<\alpha<1$. Let $t=\lfloor\frac{g-2}{2}\rfloor$ and observe that for
$G[B_{t}(v)]$ is a tree. For sufficiently large $q$ there exists an
$0<\alpha<1$ such that

$\displaystyle|B_{t}(v)\setminus B_{t-1}(v)|=d(d-1)^{t-1}=(1-o(1))d^{t}\ll
n^{1-\alpha}.$

Let such an $\alpha$ be fixed. Then, for sufficiently large $q$, Lemma 3.1
implies that for any $S\subset V$ with $|S|\ll n^{1-\alpha}$,

$\displaystyle|B_{t+1}(S)|\geq|S|(1+(1-o(1))\frac{d}{4})^{t+1}=|S|(\frac{d}{4+o(1)})^{t+1}.$

Now let $C$ be a random subset of $V$ such that for all $u\in V$,
$\mathbb{P}(u\in C)=4\varepsilon t(\frac{d}{4+o(1)})^{1-t}$ where
$\varepsilon\geq\frac{4+o(1)}{d}$. For a fixed $v\in V$ and let $H_{v}$ be the
random bipartite graph formed from $G$ and $C$ as follows; take
$V(H_{v})=(B_{t}(v)\setminus B_{t-1}(v))\cup(C\cap B_{t+1}(B_{t}(v)\setminus
B_{t-1}(v))$ where $B_{t}(v)\setminus B_{t-1}(v)$ and $C\cap
B_{t+1}(B_{t}(v)\setminus B_{t-1}(v)$ is the bipartition, then let there be an
edge from $u\in B_{t}(v)\setminus B_{t-1}(v)$ to $w\in C\cap
B_{t+1}(B_{t}(v)\setminus B_{t-1}(v)$ iff $w\in B_{t+1}(u)$. Let $E_{v}$ be
the event that there is no $(B_{t}(v)\setminus B_{t-1}(v))$-saturating
matching in $H_{v}$.

For fixed $S\subset V$ such that $|S|\ll n^{1-\alpha}$ let $A_{S}$ be the
random variable that counts $|C\cap B_{t+1}(S)|$. Then $A_{S}$ is a binomial
random variable, hence, $\mathbb{E}(A_{S})\geq(1-o(1))|S|4\varepsilon
t\frac{d^{2}}{16+o(1)}\geq(4-o(1))|S|t\frac{d}{4+o(1)}$, and applying
Chernoff’s Bound,

$\displaystyle\mathbb{P}(A_{S}\leq\mathbb{E}(A_{S})+k)\leq\operatorname{exp}(-\frac{k^{2}}{2\mathbb{E}(A_{S})}).$

Observe that applying Hall’s Marriage Theorem and the union bound, and letting
$k=(2-o(1))|S|t\frac{d}{4+o(1)}$ and $|S|=s$

$\displaystyle\mathbb{P}(E_{v})\leq\sum_{m=1}^{d^{t}}\dbinom{d^{t}}{m}\mathbb{P}(A_{S}<s\mid
s=m)$
$\displaystyle<\sum_{m=1}^{d^{t}}\dbinom{d^{t}}{m}\mathbb{P}(A_{S}\leq\frac{1}{2}\mathbb{E}(A_{S}))$
$\displaystyle\leq\sum_{m=1}^{d^{t}}\dbinom{d^{t}}{m}\mathbb{P}(A_{S}\leq(2-o(1))|S|t\frac{d}{4+o(1)})$
$\displaystyle\leq\sum_{m=1}^{d^{t}}d^{tm}\operatorname{exp}(-\frac{k^{2}}{2\mathbb{E}(A_{S})})$
$\displaystyle\leq\sum_{m=1}^{d^{t}}\operatorname{exp}(\ln(d)mt-(1-o(1))mt\frac{d}{4+o(1)})<R$

as $q,p\rightarrow\infty$ and therefore $t,d\rightarrow\infty$ where $R<1$ is
a constant, by the convergence of geometric series. Assume $\sqrt{p}\leq q\leq
e^{p}$. Finally, consider that when $q\leq e^{p+1}$,

$\displaystyle\mathbb{P}(\bigcup_{v\in V}E_{v})\leq n\mathbb{P}(E_{v})$
$\displaystyle\leq
n\sum_{m=1}^{d^{t}}\operatorname{exp}(\ln(d)mt-(1-o(1))mt\frac{d}{4+o(1)})$
$\displaystyle\leq(1+o(1))e^{3p+3}\sum_{m=1}^{d^{t}}\operatorname{exp}(\ln(d)mt-(1-o(1))mt\frac{d}{4+o(1)})$
$\displaystyle=\sum_{m=1}^{d^{t}}\operatorname{exp}(3d+\ln(d)mt-(1-o(1))mt\frac{d}{4+o(1)})<N$

as $q,p\rightarrow\infty$ and therefore $t,d\rightarrow\infty$ where $N<1$ is
a constant, again given the convergence of geometric series.

So if $q\leq e^{p+1}\rightarrow\infty$, for every $v\in V$, $H_{v}$ has a
$(B_{t}(v)\setminus B_{t-1}(v))$-saturating matching with probability bounded
away from $0$. Of course the existence of such a matching implies that if we
place cops on $C$ at the beginning of the game, then cops win with probability
bounded away from $0$. This is because if the robber begins at $v$, then the
cops win if for all $u\in B_{t}(v)\setminus B_{t-1}(v)$ and $(u,w)$ in a
$(B_{t}(v)\setminus B_{t-1}(v))$-saturating matching of $H_{v}$ the cop at $w$
takes a shortest path to $u$, then there will be a cop on every vertex of
$B_{t}(v)\setminus B_{t-1}(v)$ within $t+1$ cop turns. Hence, by the same
argument as in Theorem 1.1 the robber will be captured in $2t+1$ turns. Thus,
cops win starting on $C$ with probability bounded away from $0$ as
$p,q\rightarrow\infty$.

As in Theorem 1.1 we proceed to bound the size of $C$ using Chernoff’s Bound.
Observe that,

$\displaystyle\mathbb{E}(|C|)=n4\varepsilon t(\frac{d}{4+o(1)})^{1-t}$
$\displaystyle\leq(2\varepsilon+o(1))g(d-1)^{(\frac{3}{4}+o(1))g}(\frac{d}{4+o(1)})^{1-t}$

given $n=(1-o(1))q^{3}$ and $g\geq 4\log_{p}(q)$ implies
$n\leq(d-1)^{(1+o(1))\frac{3}{4}g}$. From this,

$\displaystyle\mathbb{E}(|C|)\leq(2\varepsilon+o(1))g(d-1)^{(3+\log_{p}(4+o(1))-2+o(1))\frac{g}{4}}$
$\displaystyle=(2\varepsilon+o(1))g(d-1)^{(1+o(1))\frac{g}{4}}=o(g(d-1)^{(1+o(1))\frac{g}{4}})$

as $p,q\rightarrow\infty$ given we can pick $\varepsilon\rightarrow 0$ due to
our only assumption regarding $\varepsilon$ be that it be great then or equal
to $\frac{4+o(1)}{d}\rightarrow 0$. From here it is easy to see that using the
same Chernoff Bound argument as in the proof of Theorem 1.1 that

$\displaystyle\mathbb{P}(|C|\leq(1+o(1))\mathbb{E}(|C|))\rightarrow 1$

as $p,q\rightarrow\infty$. This concludes the proof. ∎

##  4\. Conclusion

In this paper we have improved an upper bound for the cop number given by
Frankl appearing in [15] for of graphs with girth at least $7$ and large
minimum degree. In doing so we have shown that such graphs have cop number
$o(n\delta^{(1-o(1))\frac{g}{4}})$ where the asymptotics here are in $n$. This
result implies that for all $\alpha\in(0,1)$ then there exists a sufficiently
large $0<\varepsilon<1$ such that if $G$ has $\delta\geq n^{\varepsilon}$ and
has at most $O(n^{1-\alpha})$ cycles of length $6$ or less, then
$c(G)=O(n^{1-\alpha})$. We conjecture that this upper bound is not best
possible, as the the capture time implied by our proof is independent of the
size of our graphs, which leads us to believe that there may be more
complicated strategy which proves more advantageous for the cops. We also
conjecture that our bound holds for graphs with girth less than $7$ and graphs
with minimum degree $\delta=O(\ln{n})$ and less. Additionally, we have proven
a conjecture of Bradshaw, Hosseini, Mohar, and Stacho from [10], and in so
doing shown that the lower bound
$c(G)\geq\frac{1}{g}(\delta-1)^{\lfloor\frac{g-1}{4}\rfloor}$ is within a sub-
exponential factor of being tight.

## Acknowledgements

We would also like to acknowledge the support of the Natural Sciences and
Engineering Research Council of Canada (NSERC) for support through the
Canadian Graduate Scholarship – Master’s program.

## References

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##  Appendix A Radial velocity tables

Tables 8 to 10 list the radial velocities used in this work. For stars with
observations from different instruments, RV from each one are labelled as C
(CARMENES), H (HARPS), F (FEROS), and U (UVES). For SB1 systems in Table 8,
the methods used to obtain the RV values are labelled as T (todmor) and S
(serval).

Table 8: RVs of SB1s. Obs. date | RV | Inst. | Method
---|---|---|---
[BJD] | [m s-1]
GJ 207.1
2453708.8474 | $-3700\pm 1200$ | F | T
2453424.6545 | $55250\pm 940$ | F | T
2454455.6240 | $-2810\pm 260$ | H | T
2457693.6299 | $64069\pm 12$ | C | S
2457985.6872 | $11450\pm 43$ | C | S
2458007.6799 | $29835.5\pm 9.6$ | C | S
2458026.6315 | $57691.9\pm 8.5$ | C | S
2458078.5036 | $65398\pm 15$ | C | S
2458094.5798 | $-67.0\pm 9.4$ | C | S
2458160.3600 | $-2658\pm 17$ | C | S
2458167.3276 | $65932\pm 19$ | C | S
2458173.2985 | $54835\pm 19$ | C | S
2458182.3093 | $38259\pm 16$ | C | S
2458187.4240 | $31455\pm 35$ | C | S
2458206.3402 | $-1822\pm 15$ | C | S
2458434.5484 | $24896\pm 12$ | C | S
2458435.7075 | $42661.4\pm 9.4$ | C | S
GJ 912
2455046.8642 | $-1016.2\pm 3.2$ | H | S
2455053.8299 | $-968.7\pm 1.4$ | H | S
2455054.8321 | $-965.6\pm 1.0$ | H | S
2455056.8260 | $-931.5\pm 6.0$ | H | S
2455056.9116 | $-952.9\pm 1.9$ | H | S
2455057.8606 | $-948.9\pm 2.9$ | H | S
2455125.6477 | $-568.2\pm 1.2$ | H | S
2455464.6903 | $65.9\pm 1.0$ | H | S
2456235.5404 | $2.2\pm 1.2$ | H | S
2456502.9029 | $815\pm 447$ | F | T
2456612.6329 | $507\pm 396$ | F | T
2456617.7031 | $749\pm 362$ | F | T
2457593.6588 | $288.8\pm 2.0$ | C | S
2457595.6683 | $283.3\pm 2.0$ | C | S
2457604.6409 | $283.5\pm 2.4$ | C | S
2457608.6448 | $283.1\pm 2.1$ | C | S
2457625.6036 | $276.8\pm 2.2$ | C | S
2457634.5720 | $265.4\pm 2.4$ | C | S
2457644.5412 | $263.2\pm 2.3$ | C | S
2457672.5148 | $257.9\pm 1.7$ | C | S
2457753.2839 | $228.4\pm 2.2$ | C | S
2457931.6397 | $137.9\pm 2.2$ | C | S
2457951.6217 | $129.5\pm 2.2$ | C | S
2457971.5873 | $123.9\pm 1.8$ | C | S
2457997.5621 | $110.2\pm 1.6$ | C | S
2458017.4397 | $105.4\pm 1.9$ | C | S
2458065.4532 | $79.4\pm 1.6$ | C | S
2458095.3024 | $65.2\pm 1.5$ | C | S
2458134.2945 | $45.3\pm 1.6$ | C | S
2458289.6531 | $-30.1\pm 1.7$ | C | S
2458349.6060 | $-57.8\pm 1.1$ | C | S
2458416.4309 | $-95.7\pm 1.2$ | C | S
2458476.3298 | $-133.2\pm 1.3$ | C | S
2458663.6613 | $-242.1\pm 1.9$ | C | S
2458723.5831 | $-282.1\pm 2.1$ | C | S
2458783.4458 | $-326.4\pm 1.8$ | C | S
2458844.2613 | $-373.0\pm 3.6$ | C | S
2459033.6486 | $-528.8\pm 2.2$ | C | S
2459093.5803 | $-590.4\pm 2.1$ | C | S
2459153.4087 | $-649.2\pm 2.6$ | C | S
Table 8: Continued. Obs. date | RV | Inst. | Method
---|---|---|---
[BJD] | [m s-1]
GJ 3626
2457472.5069 | $-1426.4\pm 1.7$ | C | S
2457534.3920 | $-1489.2\pm 2.6$ | C | S
2458078.6861 | $-604.4\pm 1.3$ | C | S
2458092.6519 | $-532.8\pm 1.5$ | C | S
2458110.6715 | $-442.8\pm 1.6$ | C | S
2458112.6498 | $-431.5\pm 1.3$ | C | S
2458161.5900 | $-210.8\pm 5.6$ | C | S
2458205.4683 | $-47.6\pm 1.4$ | C | S
2458236.5044 | $35.5\pm 1.8$ | C | S
2458270.4148 | $119.9\pm 1.7$ | C | S
2458300.3645 | $166.2\pm 1.5$ | C | S
2458413.7056 | $260.2\pm 1.3$ | C | S
2458449.7141 | $272.8\pm 3.6$ | C | S
2458480.6987 | $265.3\pm 1.4$ | C | S
2458510.6934 | $268.0\pm 1.6$ | C | S
2458540.7452 | $253.1\pm 1.8$ | C | S
2458570.5584 | $246.2\pm 1.8$ | C | S
2458600.4789 | $229.8\pm 1.1$ | C | S
2458608.5203 | $222.0\pm 1.5$ | C | S
2458615.4695 | $219.8\pm 1.4$ | C | S
2458626.4686 | $219.4\pm 1.4$ | C | S
2458633.4138 | $212.6\pm 2.3$ | C | S
2458641.4210 | $210.4\pm 1.6$ | C | S
2458649.3983 | $206.3\pm 1.9$ | C | S
2458656.3797 | $199.9\pm 1.2$ | C | S
2458663.3722 | $196.6\pm 2.2$ | C | S
2458664.3700 | $193.8\pm 1.4$ | C | S
2458804.7036 | $100.5\pm 2.6$ | C | S
2458811.7346 | $91.4\pm 2.8$ | C | S
2458831.7261 | $82.0\pm 2.4$ | C | S
2458844.7072 | $67.9\pm 3.1$ | C | S
2458851.6835 | $63.0\pm 1.3$ | C | S
2458860.6576 | $58.2\pm 1.7$ | C | S
2458878.6207 | $44.3\pm 2.2$ | C | S
2458885.6051 | $34.1\pm 1.6$ | C | S
2458893.6085 | $28.4\pm 1.2$ | C | S
2458903.5747 | $18.4\pm 1.4$ | C | S
2458911.6684 | $20.8\pm 4.4$ | C | S
2458918.5292 | $9.1\pm 1.9$ | C | S
2458978.3929 | $-37.3\pm 3.1$ | C | S
2459149.6687 | $-174.1\pm 2.3$ | C | S
2459212.7667 | $-231.5\pm 3.8$ | C | S
LSPM 427-016
2456353.6532 | $12860\pm 130$ | F | T
2456353.6711 | $12830\pm 240$ | F | T
2457449.5866 | $-202.2\pm 1.8$ | C | S
2457673.7141 | $-160.3\pm 1.7$ | C | S
2458059.6730 | $-73.3\pm 1.9$ | C | S
2458091.6469 | $-72.4\pm 1.3$ | C | S
2458110.4906 | $-66.0\pm 1.5$ | C | S
2458121.5078 | $-72.1\pm 1.4$ | C | S
2458135.6016 | $-62.1\pm 1.9$ | C | S
2458166.5454 | $-54.1\pm 1.3$ | C | S
2458209.4565 | $-47.6\pm 1.9$ | C | S
2458265.3527 | $-34.3\pm 1.7$ | C | S
2458405.6881 | $3.4\pm 6.0$ | C | S
2458435.7292 | $7.8\pm 1.7$ | C | S
2458468.6965 | $11.9\pm 1.8$ | C | S
2458498.6532 | $20.0\pm 1.4$ | C | S
2458528.5540 | $25.2\pm 1.2$ | C | S
2458558.4795 | $33.0\pm 1.5$ | C | S
2458589.3391 | $38.1\pm 1.4$ | C | S
2458631.3536 | $51.8\pm 1.6$ | C | S
Table 8: Continued. Obs. date | RV | Inst. | Method
---|---|---|---
[BJD] | [m s-1]
2458769.6981 | $76.8\pm 2.1$ | C | S
2458801.7426 | $94.5\pm 2.3$ | C | S
2459151.6447 | $180.0\pm 1.3$ | C | S
2459183.7294 | $187.5\pm 1.4$ | C | S
GJ 282 C
2454040.7842 | $-20540\pm 210$ | F | T
2454040.8059 | $-20490\pm 200$ | F | T
2456294.6551 | $-16250\pm 190$ | F | T
2456982.8285 | $2255.3\pm 1.8$ | H | S
2456982.8190 | $2257.7\pm 1.9$ | H | S
2456986.8692 | $2238.7\pm 1.4$ | H | S
2456987.8191 | $2231.3\pm 1.7$ | H | S
2456987.8289 | $2233.0\pm 1.5$ | H | S
2456988.8492 | $2209.4\pm 1.8$ | H | S
2457342.8600 | $2016.3\pm 1.4$ | H | S
2457342.8709 | $2015.4\pm 1.7$ | H | S
2457344.8348 | $2010.5\pm 1.7$ | H | S
2457344.8453 | $2010.3\pm 1.6$ | H | S
2457395.5526 | $898.9\pm 2.7$ | C | S
2457401.5286 | $874.4\pm 1.8$ | C | S
2457444.4238 | $842.0\pm 2.0$ | C | S
2457466.4221 | $828.7\pm 2.1$ | C | S
2457489.3509 | $804.0\pm 2.6$ | C | S
2457493.4898 | $1903.3\pm 1.4$ | H | S
2457493.5007 | $1901.8\pm 1.4$ | H | S
2457494.5592 | $1885.5\pm 1.4$ | H | S
2457494.5699 | $1884.9\pm 1.4$ | H | S
2457699.7112 | $590.1\pm 3.4$ | C | S
2457704.6305 | $573.3\pm 2.1$ | C | S
2457712.7522 | $1645.8\pm 1.7$ | H | S
2457712.7627 | $1642.7\pm 1.6$ | H | S
2457760.5315 | $500.0\pm 2.4$ | C | S
2457769.6272 | $1587.7\pm 1.0$ | H | S
2457769.6380 | $1589.4\pm 1.0$ | H | S
2457771.7639 | $1596.6\pm 1.8$ | H | S
2457771.7745 | $1599.5\pm 1.6$ | H | S
2457788.5567 | $488.4\pm 2.4$ | C | S
2457791.4466 | $421.5\pm 3.7$ | C | S
2457793.4649 | $457.6\pm 2.3$ | C | S
2457800.4426 | $483.2\pm 2.9$ | C | S
2457821.4008 | $420.9\pm 1.9$ | C | S
2457853.3705 | $357.7\pm 2.0$ | C | S
2457857.3579 | $374.3\pm 2.3$ | C | S
2458025.7141 | $116.4\pm 1.9$ | C | S
2458029.6965 | $131.0\pm 2.7$ | C | S
2458034.6884 | $90.9\pm 2.3$ | C | S
2458040.6785 | $91.7\pm 2.7$ | C | S
2458048.6727 | $82.8\pm 2.0$ | C | S
2458052.6611 | $66.0\pm 1.9$ | C | S
2458054.7226 | $78.1\pm 2.2$ | C | S
2458056.6422 | $66.2\pm 1.4$ | C | S
2458059.6546 | $50.3\pm 2.1$ | C | S
2458065.5992 | $48.7\pm 1.5$ | C | S
2458074.6706 | $41.2\pm 1.7$ | C | S
2458081.5932 | $6.6\pm 2.4$ | C | S
2458091.5506 | $1.7\pm 1.9$ | C | S
2458094.5913 | $-20.9\pm 1.8$ | C | S
2458097.7185 | $-5.0\pm 2.3$ | C | S
2458123.4250 | $-48.3\pm 1.8$ | C | S
2458140.3853 | $-83.3\pm 2.0$ | C | S
2458166.4032 | $-147.2\pm 2.5$ | C | S
2458200.3147 | $-167.9\pm 2.2$ | C | S
2458382.7032 | $-565.3\pm 2.5$ | C | S
Table 8: Continued. Obs. date | RV | Inst. | Method
---|---|---|---
[BJD] | [m s-1]
2458451.6149 | $-704.8\pm 2.0$ | C | S
2458518.5736 | $-889.2\pm 2.6$ | C | S
2458524.5624 | $-900.0\pm 2.2$ | C | S
2458526.5304 | $211.3\pm 1.5$ | H | S
2458526.5410 | $212.6\pm 1.5$ | H | S
2458527.5310 | $226.9\pm 1.5$ | H | S
2458527.5574 | $224.3\pm 1.2$ | H | S
2458532.5496 | $152.2\pm 1.1$ | H | S
2458532.5769 | $152.1\pm 1.2$ | H | S
2458533.5557 | $181.4\pm 1.0$ | H | S
2458533.5807 | $180.4\pm 1.0$ | H | S
2458534.5343 | $198.3\pm 1.1$ | H | S
2458534.5614 | $198.3\pm 1.1$ | H | S
2458585.3259 | $-1027.5\pm 2.1$ | C | S
2458592.4805 | $12.6\pm 1.6$ | H | S
2458592.5159 | $11.3\pm 1.7$ | H | S
2458595.4924 | $45.6\pm 1.1$ | H | S
2458595.5258 | $44.6\pm 1.2$ | H | S
2458606.4830 | $-0.2\pm 0.9$ | H | S
2458606.5131 | $-1.2\pm 0.8$ | H | S
2458607.4725 | $7.6\pm 0.9$ | H | S
2458607.4982 | $9.5\pm 0.9$ | H | S
2458758.7044 | $-1461.6\pm 2.2$ | C | S
2458836.6590 | $-1687.4\pm 7.6$ | C | S
2458896.4318 | $-1869.8\pm 2.0$ | C | S
2459132.7130 | $-2539.1\pm 3.4$ | C | S
Table 9: RVs of SB2s.

Obs. date | RVA | RVB
---|---|---
$[$BJD$]$ | [km s-1] | [km s-1]
UCAC4 355-020729
2457695.6848 | $13.294\pm 0.062$ | $14.909\pm 0.094$
2457759.5452 | $11.756\pm 0.050$ | $16.571\pm 0.075$
2457788.4644 | $15.282\pm 0.049$ | $12.382\pm 0.077$
2457799.4012 | $11.619\pm 0.041$ | $16.650\pm 0.069$
2457824.3610 | $12.993\pm 0.047$ | $15.180\pm 0.071$
2457832.3566 | $11.779\pm 0.043$ | $16.532\pm 0.062$
2458034.7061 | $12.218\pm 0.047$ | $16.072\pm 0.077$
2458043.7118 | $14.697\pm 0.045$ | $12.940\pm 0.073$
2458047.7093 | $12.394\pm 0.039$ | $15.835\pm 0.064$
2458051.7069 | $15.888\pm 0.039$ | $11.804\pm 0.061$
2458054.7050 | $11.699\pm 0.042$ | $16.507\pm 0.069$
2458058.6955 | $15.850\pm 0.047$ | $11.804\pm 0.075$
2458061.7292 | $11.578\pm 0.084$ | $16.671\pm 0.135$
2458064.6885 | $15.795\pm 0.059$ | $11.754\pm 0.096$
2458074.6442 | $11.563\pm 0.041$ | $16.687\pm 0.062$
2458078.6349 | $15.687\pm 0.040$ | $11.895\pm 0.067$
2458084.6432 | $15.848\pm 0.051$ | $11.732\pm 0.082$
2458123.5290 | $15.697\pm 0.043$ | $11.873\pm 0.071$
2458167.4181 | $12.518\pm 0.043$ | $15.629\pm 0.069$
2458172.3816 | $12.309\pm 0.051$ | $15.911\pm 0.081$
2458173.3962 | $11.710\pm 0.042$ | $16.525\pm 0.072$
2458174.3848 | $13.157\pm 0.053$ | $15.019\pm 0.086$
2458175.3863 | $15.206\pm 0.049$ | $12.505\pm 0.077$
2458182.3797 | $15.587\pm 0.065$ | $12.055\pm 0.099$
2458191.3431 | $13.968\pm 0.031$ | $\cdots$
2458426.6586 | $15.395\pm 0.047$ | $12.300\pm 0.073$
2458433.6589 | $14.877\pm 0.046$ | $12.815\pm 0.076$
UU UMi
2456418.4924aa$a$RV from a CAFE spectrum. | $-41.365\pm 0.162$ | $\cdots$
2457472.6413 | $-39.007\pm 0.035$ | $-45.157\pm 0.141$
2457504.5670 | $-38.904\pm 0.038$ | $-45.141\pm 0.147$
2457529.4912 | $-38.968\pm 0.037$ | $-45.298\pm 0.141$
2457556.4629 | $-38.860\pm 0.041$ | $-45.265\pm 0.161$
2457559.5434 | $-38.858\pm 0.043$ | $-45.269\pm 0.174$
2457763.6439 | $-38.781\pm 0.039$ | $-45.590\pm 0.148$
2457800.7530 | $-38.690\pm 0.025$ | $-45.638\pm 0.111$
2457815.5392 | $-38.692\pm 0.030$ | $-45.635\pm 0.135$
2457832.5606 | $-38.701\pm 0.025$ | $-45.645\pm 0.122$
2457848.6126 | $-38.663\pm 0.027$ | $-45.662\pm 0.115$
2457867.5390 | $-38.681\pm 0.027$ | $-45.678\pm 0.133$
2457897.4662 | $-38.693\pm 0.031$ | $-45.680\pm 0.135$
2457931.5387 | $-38.677\pm 0.033$ | $-45.713\pm 0.152$
2457961.3985 | $-38.612\pm 0.029$ | $-45.732\pm 0.142$
2457993.3479 | $-38.677\pm 0.036$ | $-45.774\pm 0.166$
2458054.3032 | $-38.614\pm 0.029$ | $-45.769\pm 0.122$
2458117.7403 | $-38.643\pm 0.030$ | $-45.772\pm 0.122$
2458161.7033 | $-38.647\pm 0.033$ | $-45.784\pm 0.131$
2458200.5577 | $-38.666\pm 0.034$ | $-45.763\pm 0.143$
2458263.4946 | $-38.681\pm 0.031$ | $-45.727\pm 0.133$
2458353.3698 | $-38.665\pm 0.033$ | $-45.661\pm 0.145$
2458451.7424 | $-38.766\pm 0.029$ | $-45.599\pm 0.127$
2458541.6890 | $-38.788\pm 0.029$ | $-45.484\pm 0.120$
2458631.5092 | $-38.895\pm 0.030$ | $-45.274\pm 0.138$
2458723.3696 | $-38.985\pm 0.029$ | $-45.091\pm 0.136$
2458816.7528 | $-39.113\pm 0.028$ | $-44.931\pm 0.129$
2458913.6613 | $-39.255\pm 0.028$ | $-44.616\pm 0.121$
2459003.4687 | $-39.474\pm 0.035$ | $-44.412\pm 0.144$
2459093.3467 | $-39.692\pm 0.029$ | $-44.050\pm 0.144$
777

Table 10: RVs of ST3s. Obs. date | RVA | RVBa | RVBb | Inst.
---|---|---|---|---
$[$BJD$]$ | [km s-1] | [km s-1] | [km s-1]
GJ 3916
2453099.8240 | $4.073(95)$ | $-10.21(41)$ | $12.11(33)$ | U
2453112.6457 | $4.25(12)$ | $\cdots$ | $\cdots$ | U
2453453.7944 | $8.291(97)$ | $\cdots$ | $\cdots$ | U
2453480.6825 | $8.879(97)$ | $-18.55(41)$ | $\cdots$ | U
2453488.6672 | $8.736(92)$ | $-18.63(36)$ | $\cdots$ | U
2453505.8286 | $8.31(10)$ | $-8.90(37)$ | $2.58(25)$ | U
2454167.8224 | $3.18(95)$ | $-5.0(3.3)$ | $11.50(1.84)$ | F
2454569.6487 | $-1.675(99)$ | $\cdots$ | $14.34(30)$ | F
2454570.8108 | $-2.037(86)$ | $\cdots$ | $12.84(27)$ | H
2454580.8485 | $-1.24(10)$ | $\cdots$ | $\cdots$ | F
2454583.8633 | $-2.029(84)$ | $11.57(25)$ | $3.25(26)$ | H
2454628.5259 | $-2.43(72)$ | $16.9(2.5)$ | $\cdots$ | F
2454639.7481 | $-1.83(70)$ | $13.2(2.5)$ | $\cdots$ | F
2454661.5681 | $-2.248(79)$ | $\cdots$ | $14.54(29)$ | H
2454939.7950 | $-2.616(82)$ | $-9.26(21)$ | $20.77(26)$ | F
2454943.8549 | $-2.544(96)$ | $-9.67(26)$ | $22.06(30)$ | F
2454951.8029 | $-2.623(91)$ | $-9.71(24)$ | $22.11(30)$ | F
2454984.6058 | $-0.876(82)$ | $13.04(31)$ | $\cdots$ | F
2454993.7310 | $-1.771(80)$ | $17.20(32)$ | $\cdots$ | F
2455003.5563 | $-2.36(14)$ | $20.80(48)$ | $\cdots$ | F
2455053.5303 | $-0.846(87)$ | $\cdots$ | $9.75(35)$ | F
2455066.4967 | $-1.995(93)$ | $\cdots$ | $17.59(28)$ | F
2455075.4993 | $-2.10(70)$ | $\cdots$ | $20.9(2.4)$ | F
2455338.7519 | $-0.91(10)$ | $-8.80(25)$ | $19.37(28)$ | F
2455397.5650 | $-0.798(88)$ | $18.45(29)$ | $-7.40(17)$ | F
2455407.5645 | $-1.127(86)$ | $19.65(32)$ | $-7.97(20)$ | F
2455430.5447 | $-0.392(81)$ | $14.24(33)$ | $\cdots$ | F
2455434.5660 | $-0.20(75)$ | $12.5(2.3)$ | $\cdots$ | F
2455438.4973 | $0.14(64)$ | $\cdots$ | $\cdots$ | F
2455615.8289 | $0.77(10)$ | $-11.16(34)$ | $19.57(26)$ | F
2456352.8951 | $6.87(75)$ | $\cdots$ | $-10.2(2.4)$ | F
2456500.6423 | $7.59(91)$ | $\cdots$ | $-6.0(2.1)$ | F
2457476.6461 | $-1.098(78)$ | $-8.69(22)$ | $22.32(24)$ | C
2457493.6138 | $-1.448(34)$ | $-1.22(29)$ | $14.76(22)$ | C
2457504.5889 | $-1.511(43)$ | $6.67(25)$ | $7.70(18)$ | C
2457540.4868 | $-2.167(41)$ | $21.32(27)$ | $-4.59(37)$ | C
2457910.4391 | $-1.398(60)$ | $14.35(29)$ | $1.48(48)$ | C
2457916.5426 | $-1.794(47)$ | $17.79(34)$ | $-1.61(36)$ | C
2457949.3780 | $-1.865(42)$ | $18.61(32)$ | $-1.99(34)$ | C
2457977.3697 | $-1.537(36)$ | $3.81(34)$ | $11.21(26)$ | C
2458166.6713 | $-1.126(35)$ | $6.91(55)$ | $7.00(42)$ | C
2458177.7400 | $-0.872(62)$ | $14.23(33)$ | $-0.63(36)$ | C
2458186.7186 | $-1.501(40)$ | $18.85(27)$ | $-2.95(38)$ | C
2458200.6650 | $-1.417(41)$ | $20.75(22)$ | $-5.71(23)$ | C
2458207.6529 | $-1.469(41)$ | $19.72(23)$ | $-4.83(29)$ | C
2458225.6859 | $-0.626(49)$ | $13.26(28)$ | $0.26(36)$ | C
2458237.5692 | $-0.749(65)$ | $6.68(35)$ | $6.63(27)$ | C
2458244.5965 | $-0.544(38)$ | $1.02(55)$ | $10.43(27)$ | C
2458249.6074 | $-0.835(33)$ | $-0.65(33)$ | $13.27(24)$ | C
2458261.4581 | $-0.905(37)$ | $-7.11(25)$ | $19.01(23)$ | C
2458269.5321 | $-0.803(35)$ | $-9.38(25)$ | $21.28(22)$ | C
2458275.5228 | $-0.789(56)$ | $-9.26(30)$ | $21.09(26)$ | C
2458283.5380 | $-0.813(46)$ | $-6.85(28)$ | $18.48(24)$ | C
2458289.5200 | $-0.921(38)$ | $-1.68(47)$ | $14.87(26)$ | C
2458297.4920 | $-0.390(42)$ | $4.63(26)$ | $8.82(18)$ | C
2458575.7001 | $0.331(53)$ | $12.69(25)$ | $-0.46(32)$ | C
2458587.6266 | $0.657(37)$ | $18.01(22)$ | $-6.07(21)$ | C
2458597.6156 | $0.725(62)$ | $19.26(29)$ | $-7.15(25)$ | C
2458603.5602 | $0.753(38)$ | $18.55(24)$ | $-6.79(22)$ | C
2458609.5737 | $0.732(42)$ | $17.15(23)$ | $-5.63(21)$ | C
2458615.5395 | $0.446(38)$ | $15.19(25)$ | $-4.18(26)$ | C
2458626.5359 | $0.830(46)$ | $10.28(31)$ | $0.60(31)$ | C
2458636.4799 | $1.381(35)$ | $5.70(16)$ | $5.74(14)$ | C
Table 10: Continued.

Obs. date | RVA | RVBa | RVBb | Inst.
---|---|---|---|---
$[$BJD$]$ | [km s-1] | [km s-1] | [km s-1]
2458643.4509 | $0.894(35)$ | $0.64(35)$ | $9.36(26)$ | C
2458656.4064 | $0.993(39)$ | $-7.21(25)$ | $15.64(22)$ | C
2458662.3779 | $1.102(43)$ | $-9.81(26)$ | $17.89(25)$ | C
2458679.3749 | $1.157(39)$ | $-9.89(25)$ | $17.94(22)$ | C
2458690.3681 | $1.000(38)$ | $-2.14(43)$ | $11.34(28)$ | C
2458696.3722 | $1.774(60)$ | $-0.16(31)$ | $7.22(30)$ | C
2458708.3595 | $0.956(40)$ | $12.13(30)$ | $-0.72(44)$ | C
GJ 4383
2455046.8563 | $-12.63(11)$ | $\cdots$ | $\cdots$ | H
2455053.8419 | $-12.608(96)$ | $\cdots$ | $\cdots$ | H
2455054.8560 | $-12.622(87)$ | $\cdots$ | $\cdots$ | H
2455056.8369 | $-12.58(12)$ | $\cdots$ | $\cdots$ | H
2455056.8994 | $-12.58(11)$ | $\cdots$ | $\cdots$ | H
2455124.6171 | $-12.257(83)$ | $\cdots$ | $\cdots$ | H
2455537.5585 | $-10.154(87)$ | $\cdots$ | $\cdots$ | H
2455541.5532 | $-10.067(85)$ | $\cdots$ | $\cdots$ | H
2455550.5263 | $-9.997(94)$ | $\cdots$ | $\cdots$ | H
2456496.8926 | $-7.47(28)$ | $\cdots$ | $\cdots$ | F
2456496.9060 | $-7.49(22)$ | $\cdots$ | $\cdots$ | F
2456614.6481 | $-7.54(22)$ | $\cdots$ | $\cdots$ | F
2456614.6626 | $-7.58(22)$ | $\cdots$ | $\cdots$ | F
2457587.6611 | $-10.559(38)$ | $\cdots$ | $\cdots$ | C
2457604.6500 | $-10.629(30)$ | $\cdots$ | $\cdots$ | C
2457622.6367 | $-10.730(29)$ | $\cdots$ | $-18.93(53)$ | C
2457623.5259 | $-10.57(31)$ | $\cdots$ | $\cdots$ | C
2457623.5458 | $-10.717(33)$ | $-6.08(75)$ | $-19.17(50)$ | C
2457654.5045 | $-10.941(31)$ | $-0.46(48)$ | $-24.88(53)$ | C
2457672.4896 | $-10.984(29)$ | $-1.26(40)$ | $-23.64(45)$ | C
2457695.3885 | $-11.147(33)$ | $\cdots$ | $\cdots$ | C
2457786.2850 | $-11.615(43)$ | $\cdots$ | $\cdots$ | C
2457916.6483 | $-12.511(25)$ | $\cdots$ | $-6.01(58)$ | C
2457942.6396 | $-12.612(27)$ | $-19.57(42)$ | $2.28(50)$ | C
2457999.5659 | $-12.937(24)$ | $\cdots$ | $-4.17(43)$ | C
2458051.4269 | $-13.287(28)$ | $-2.49(37)$ | $\cdots$ | C
2458092.3378 | $-13.540(20)$ | $4.25(31)$ | $-20.19(35)$ | C
2458122.2522 | $-13.636(21)$ | $-4.21(36)$ | $\cdots$ | C
2458283.6381 | $-14.622(29)$ | $3.26(43)$ | $\cdots$ | C
2458343.6430 | $-14.896(22)$ | $-5.05(26)$ | $\cdots$ | C
2458410.5819 | $-15.148(33)$ | $\cdots$ | $3.99(54)$ | C
2458471.3527 | $-15.450(22)$ | $-1.05(34)$ | $-7.37(43)$ | C
2458656.6395 | $-15.975(24)$ | $-5.82(41)$ | $-0.80(57)$ | C
2458718.5956 | $-15.981(23)$ | $6.44(36)$ | $\cdots$ | C
2458784.4240 | $-16.111(25)$ | $-7.59(38)$ | $1.80(42)$ | C
2458844.3742 | $-16.019(27)$ | $-10.53(55)$ | $5.52(47)$ | C
2459074.6456 | $-15.606(23)$ | $-9.04(31)$ | $1.97(37)$ | C
2459138.4984 | $-15.398(26)$ | $2.64(35)$ | $\cdots$ | C
2459214.3691 | $-15.135(25)$ | $-8.49(38)$ | $-0.45(47)$ | C
888The number between parentheses indicate the uncertainty of the last digits.


such that $Q_{2}\preceq_{\rho}Q_{1}$. Then
$\operatorname{Pval}(Q_{1})\preceq_{\rho}\operatorname{Pval}(Q_{2})$. In
particular, if $Q_{2}\sim_{\rho}Q_{1}$, then
$\operatorname{Pval}(Q_{2})\sim_{\rho}\operatorname{Pval}(Q_{1})$.

###### Proof.

Set $\widetilde{v}_{i}=\operatorname{Pval}(Q_{i})$. Choose $\mu\geq 0$ such
that $\alpha(w)(Q_{1})\subset Q_{2}$ for any $w\in\mathbb{Z}^{n}$ with
$\rho(w)\geq\mu$ and such that there exists $w\in\mathbb{Z}^{n}$ with
$\rho(w)=\mu$. Given $h\in H$, suppose that $z\in\mathbb{Z}^{n}$ is such that
$\alpha(z)(h)\in Q_{1}$. If $w\in\mathbb{Z}^{n}$ with $\rho(w)=\rho(z)+\mu$
then we see that

$\alpha(w)(h)=\alpha(wz^{-1})(\alpha(z)(h))\in\alpha(wz^{-1})(Q_{1})\subset
Q_{2}.$

Thus $\inf\\{\rho(w):\alpha(w)(h)\in Q_{2}\\}\leq\rho(z)+\mu$. Taking the
negative infimum over all such $z$ yields
$\widetilde{v}_{2}(h)\geq\widetilde{v}_{1}(h)-\mu$, and so
$\widetilde{v}_{2}\succeq_{\rho}\widetilde{v}_{1}$, as desired. ∎

We now show that the map $\operatorname{Conf}$ is also order-reversing, and
consequently respects the equivalence relation on pseudo-valuations.

######  Lemma B.10.

If $\widetilde{v}_{1},\widetilde{v}_{2}$ are pseudo-valuations such that
$\widetilde{v}_{1}\preceq_{\rho}\widetilde{v}_{2}$, then
$\operatorname{Conf}(\widetilde{v}_{2})\preceq_{\rho}\operatorname{Conf}(\widetilde{v}_{1})$.
In particular, if $\widetilde{v}_{1}\sim_{\rho}\widetilde{v}_{2}$, then
$\operatorname{Conf}(\widetilde{v}_{1})\sim_{\rho}\operatorname{Conf}(\widetilde{v}_{2}).$

###### Proof.

Set $Q_{i}=\operatorname{Conf}(\widetilde{v}_{i})$. Since
$\widetilde{v}_{1}\preceq_{\rho}\widetilde{v}_{2}$, there exists a constant
$K$ such that $\widetilde{v}_{1}(h)\leq\widetilde{v}_{2}(h)+K$ for all $h\in
H$. It then follows from the property (PV3) applied to $\widetilde{v}_{2}$
that $\alpha(z)(Q_{1})\subset Q_{2}$ for any $z\in\mathbb{Z}^{n}$ such that
$\rho(z)\geq K$. Thus $Q_{2}\preceq_{\rho}Q_{1}$ by Lemma B.8. ∎

It follows that our maps $\operatorname{Conf}$ and $\operatorname{Pval}$
descend to maps between the equivalence classes of confining sets and
equivalence classes of pseudo-valuations subordinate to $\rho$. We will now
show that these maps are inverses, which will prove that they are order-
reversing isomorphisms. We first show the composition
$\operatorname{Pval}\circ\operatorname{Conf}$ is the identity on equivalence
classes of pseudo-valuations.

######  Proposition B.11.

If $\widetilde{v}$ is a pseudo-valuation subordinate to $\rho$, then
$\widetilde{v}\sim_{\rho}\operatorname{Pval}(\operatorname{Conf}(\widetilde{v}))$.

###### Proof.

Set $Q=\operatorname{Conf}(\widetilde{v})$ and
$\widetilde{u}=\operatorname{Pval}(Q)$. We first show that
$\widetilde{v}(h)=+\infty$ if and only if $\widetilde{u}(h)=+\infty$. Let
$\widetilde{v}(h)=+\infty$. Observe that
$\widetilde{v}(\alpha(z)(h))=\rho(z)+\widetilde{v}(h)=+\infty$ for any
$z\in\mathbb{Z}^{n}$. It follows that
$\operatorname{inf}\\{\rho(z):\alpha(z)(h)\in Q\\}=-\infty$, and thus
$\widetilde{u}(h)=+\infty$. Conversely, suppose that
$\widetilde{u}(h)=+\infty$. By the definition of $\widetilde{u}$, there is a
sequence $\\{y_{i}\\}\subset\mathbb{Z}^{n}$ with $\rho(y_{i})\to-\infty$ such
that $\alpha(y_{i})(h)\in Q$ for all $i$. But then
$\widetilde{v}(\alpha(y_{i})(h))=\rho(y_{i})+\widetilde{v}(h)\geq 0$ for all
$i$. As the $\rho(y_{i})$ are arbitrarily negative (and $\rho$ takes values
only in $\mathbb{R}$), this is only possible if $\widetilde{v}(h)=+\infty$.

We now show that there is a constant $K_{0}$ such that

$\widetilde{u}(h)\leq\widetilde{v}(h)\leq\widetilde{u}(h)+K_{0}$

for all $h\in H$, which will prove the result. Let $h\in H$. In light of the
above, we may assume that both $\widetilde{u}(h),\widetilde{v}(h)$ are finite
values, for if $\widetilde{u}(h)=+\infty=\widetilde{v}(h)$, they satisfy the
inequality for any constant $K_{0}$. Suppose that $z\in\mathbb{Z}^{n}$ such
that $\alpha(z)(h)\in Q$. By definition $\widetilde{v}(\alpha(z)(h))\geq 0$,
and therefore $\rho(z)+\widetilde{v}(h)\geq 0$ and
$\rho(z)\geq-\widetilde{v}(h).$ Taking an infimum on the left-hand side over
all such $z$ gives us that

$\operatorname{inf}\\{\rho(z):\alpha(z)(h)\in Q\\}\geq-\widetilde{v}(h).$

Thus $\widetilde{u}(h)\leq\widetilde{v}(h)$. To establish the other
inequality, note that since the image of $\rho$ in $\mathbb{R}$ is either
dense or cyclic, there exists $K_{0}>0$ such that every point of $\mathbb{R}$
is within $K_{0}$ of a point of $\operatorname{Im}(\rho)$. Choose
$z\in\mathbb{Z}^{n}$ such that $0\leq\widetilde{v}(h)+\rho(z)\leq K_{0}$. Thus
$0\leq\widetilde{v}(\alpha(z)(h))\leq K_{0}$ so that $\alpha(z)(h)\in Q$ and
$\rho(z)\leq-\widetilde{v}(h)+K_{0}$. It then follows that
$\widetilde{u}(h)\geq\widetilde{v}(h)-K_{0}$ so that
$\widetilde{v}(h)\leq\widetilde{u}(h)+K_{0}$. This completes the proof.∎

Next, we prove that the composition
$\operatorname{Conf}\circ\operatorname{Pval}$ is the identity on equivalence
classes of confining subsets.

######  Proposition B.12.

If $Q$ is confining with respect to $\rho$, then
$Q\sim_{\rho}\operatorname{Conf}(\operatorname{Pval}(Q))$.

###### Proof.

Let $\widetilde{v}=\operatorname{Pval}(Q)$ and
$Q^{\prime}=\operatorname{Conf}(\widetilde{v})$. Since $Q$ is confining with
respect to $\rho$, Definition B.2(a) yields that $q\in
Q\Rightarrow\operatorname{inf}\\{\rho(z):\alpha(z)(q)\in Q\\}\leq
0\Rightarrow\widetilde{v}(q)\geq 0\Rightarrow q\in Q^{\prime}.$ Thus $Q\subset
Q^{\prime}$ and hence $Q^{\prime}\preceq_{\rho}Q$. Conversely, let $q\in
Q^{\prime}$. Then $\widetilde{v}(q)\geq 0$, and the definition of
$\widetilde{v}$ yields that $\alpha(z)(q)\in Q$ for any $z\in\mathbb{Z}^{n}$
such that $\rho(z)>0$. Thus $Q\preceq_{\rho}Q^{\prime}$ by Lemma B.8,
completing the proof. ∎

Finally we are ready to put together the above results to prove Theorem B.1.

###### Proof of Theorem B.1.

By Lemmas B.9 and B.10, the maps $\operatorname{Conf}$ and
$\operatorname{Pval}$ descend to order-reversing maps on the equivalence
classes of pseudo-valuations subordinate to $\rho$ and confining subsets with
respect to $\rho$, respectively. It follows from Propositions B.11 and B.12
that the maps are inverses of each other on the equivalence classes. Since
both are order-reversing, they descend to inverse isomorphisms between the
poset of pseudo-valuations and the opposite of the poset of confining subsets.
∎

We end this section with a discussion of the structure of a confining subset
associated to a pseudo-valuation on $H$ subordinate to $\rho$. In particular,
we show that, up to equivalence of generating sets, $Q$ is the union of a
collection of cosets of a subgroup $K$ of $H$, defined by

$K=\\{h\in H:\widetilde{v}(h)=+\infty\\}.$

The reader may check using conditions (PV2) and (PV3) that $K$ is indeed a
subgroup. Since $Q=\\{h\in H:\widetilde{v}(h)\geq 0\\}$, it is clear that
$K\subset Q$. Let $T$ be a transversal for $K$ in $H$ containing $0$. Let
$T_{Q}\subseteq T$ be the set of representatives of cosets of $K$ that have
non-empty intersection with $Q$, and let
$\overline{K}=\displaystyle\bigcup_{g\in T_{Q}}(g+K)$.

Our goal is to prove the following.

######  Proposition B.13.

Let $\widetilde{v}$ be a pseudo-valuation on $H$ subordinate to $\rho$, and
let $Q=\operatorname{Conf}(\widetilde{v})$ be the associated confining set.
Then we have $[Q\cup Z_{\rho}]=\left[\overline{K}\cup Z_{\rho}\right].$

If $K=0$, then the proposition is vacuous. However, it may yield interesting
information if $K\neq 0$. We begin by proving some properties of the subgroup
$K$.

######  Lemma B.14.

If $\widetilde{v}$ is a pseudo-valuation on $H$ subordinate to $\rho$ and
$Q=\operatorname{Conf}(\widetilde{v})$, then $K$ is a subgroup of $H$
invariant under the action of $\mathbb{Z}^{n}$ and hence normal in $G$.
Further, $K$ is the (unique) largest such subgroup contained in $Q$.

###### Proof.

To see the invariance of $K$ under the action of $\mathbb{Z}^{n}$, observe
that $\widetilde{v}(\alpha(z)(k))=\rho(z)+\widetilde{v}(k)=+\infty$ for any
$k\in K$ and $z\in\mathbb{Z}^{n}$, as $\rho$ only takes finite values in
$\mathbb{R}$. Thus $\alpha(z)(k)\in K$ for all $k\in K,z\in\mathbb{Z}^{n}$.
The normality of $K$ follows from its invariance under $\mathbb{Z}^{n}$ and
the fact that $H$ is abelian.

To prove the final statement, suppose for contradiction that $K^{\prime}$ is a
subgroup contained in $Q$ that is invariant under the action of
$\mathbb{Z}^{n}$ and such that $K^{\prime}\backslash K\neq\emptyset$. Then
there exists an $h\in K^{\prime}$ with $0\leq\widetilde{v}(h)<+\infty$. Since
$K^{\prime}$ is $\mathbb{Z}^{n}$–invariant, $\alpha(z)(h)\in K^{\prime}$ for
any $z\in\mathbb{Z}^{n}$. However,
$\widetilde{v}(\alpha(z)(h))=\rho(z)+\widetilde{v}(h)$, and so choosing $z$
with $\rho(z)$ very negative shows there are elements in $K^{\prime}\subseteq
Q$ whose valuations are negative. This contradicts the definition of $Q$, and
so $K^{\prime}\subseteq K$. ∎

The subgroup $K$ can be very different in different cases. For instance, let
$\varphi\in SL_{3}(\mathbb{Z})$ be a matrix with irrational eigenvalues. If
$G=\mathbb{Z}^{3}\rtimes_{\varphi}\mathbb{Z}$, then $K$ has to be a
$\varphi-$invariant subgroup of $\mathbb{Z}^{3}$ for any confining subset $Q$.
If $Q$ is a strictly confining subset, we must have $K=\\{0\\}$. On the other
hand, if $Q=\mathbb{Z}^{3}$, then $K=\mathbb{Z}^{3}$. In the case of
$G=\mathbb{R}^{3}\rtimes_{\varphi}\mathbb{Z}$, the subgroup $K$ will be one of
the invariant subspaces for the action of $\varphi$ on $\mathbb{R}^{3}$,
depending on the choice of confining subset.

The next lemma shows that every strictly confining subset intersects one of
the non-trivial cosets of $K$.

######  Lemma B.15.

If $\displaystyle Q\cap\left(\bigcup_{g\in
T\setminus\\{0\\}}g+K\right)=\emptyset$, then $Q=K=H$.

###### Proof.

We must have $Q=K$, as $K\subset Q$ and $Q$ does not intersect any non-trivial
coset of $K$ by assumption. Thus $Q$ is a $\mathbb{Z}^{n}$–invariant subgroup
of $H$. By Definition B.2(b), for every $h\in H$, there is a
$z\in\mathbb{Z}^{n}$ such that $\alpha(z)(h)\in Q$, and hence
$h\in\alpha(z^{-1})(Q)=Q$.∎

We are now ready to prove Proposition B.13.

###### Proof of Proposition B.13.

We first show that $\overline{K}\cup Z_{\rho}$ is a generating set for
$G=H\rtimes_{\alpha}\mathbb{Z}^{n}$. As $Z_{\rho}$ generates $\mathbb{Z}^{n}$,
it suffices to show that $H$ is contained in the subgroup generated by
$\overline{K}\cup Z_{\rho}$.

Let $h\in H$. Then $h\in g+K$ for some $g\in T$. By Definition B.2(b), there
is a $z\in\mathbb{Z}^{n}$ such that $\alpha(z)(h)\in Q$. Hence
$\alpha(z)(h)\in Q\cap(g^{\prime}+K)$ for some $g^{\prime}\in T$. In
particular, $g^{\prime}\in T_{Q}$ and $h\in\alpha(z^{-1})(g^{\prime}+K)$, so
$h$ is contained in the subgroup generated by $\overline{K}\cup Z_{\rho}$.

To establish the equivalence of the generating sets, we will show that each
generating set has uniformly bounded word length with respect to the other.
First of all, $Q\subset\overline{K}$ by definition of $T_{Q}$.

Conversely, $K\subset Q$ by definition. If $Q$ intersects the coset $g+K$ non-
trivially, we will show that this coset has uniformly bounded word length in
$Q\cup Z_{\rho}$, which will prove the result. By assumption, there exists an
element $q\in Q$ of the form $q=g+k$ with $k\in K$. For any other
$k^{\prime}\in K$, write $g+k^{\prime}=q+(k^{\prime}-k)$. As $K$ is a subgroup
contained in $Q$, this implies that $g+k^{\prime}\in Q+Q$. If $z_{0}$ is the
element from Definition B.2(c), then $\alpha(z_{0})(g+k^{\prime})\in Q$. As
$k^{\prime}$ was arbitrary, it follows that the word length of any element of
$g+K$ with respect to $Q\cup Z_{\rho}$ is at most $2\|z_{0}\|_{Z_{\rho}}+1$. ∎

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