Cops and robbers on directed and undirected abelian Cayley graphs

TL;DR

This work analyzes the cops-and-robbers game on Cayley graphs of finite abelian groups, both directed and undirected, in the context of Meyniel's conjecture. By introducing a general inductive lemma that reduces the problem to quotient groups via a group element $k$ that accounts for many robber moves, the authors unify and extend prior methods (Frankl–Hamidoune) to obtain $O(\sqrt{n})$ bounds for both graph classes. They achieve refined upper bounds: about $0.9424\sqrt{n}+O(1)$ for undirected abelian Cayley graphs and about $1.3328\sqrt{n}+O(1)$ for directed ones, with further improvements via the smallest prime factor of $n$. Additionally, the paper constructs Meyniel extremal families with cop number $\Theta(\sqrt{n})$ using Sidon-type generating sets on $G=(\mathbb{Z}/p\mathbb{Z})^2$, and extends these constructions to all $n$ by appending paths, demonstrating the tightness of the $\Theta(\sqrt{n})$ growth in this context.

Abstract

We show that the cop number of directed and undirected Cayley graphs on abelian groups has an upper bound of the form of $O(\sqrt{n})$, where $n$ is the number of vertices, by introducing a refined inductive method. With our method, we improve the previous upper bound on cop number for undirected Cayley graphs on abelian groups, and we establish an upper bound on the cop number of directed Cayley graphs on abelian groups. We also use Cayley graphs on abelian groups to construct new \emph{Meyniel extremal families}, which contain graphs of every order $n$ with cop number $Θ(\sqrt{n})$.

Figures (2)

• Figure 1: The figure shows a cop guarding a robber move in an abelian Cayley graph. The robber's vertex is labelled $r$, and each arc is labelled with its corresponding generating element. The values $a$ and $b$ are generating elements, and $a - b = k$. Here, a cop occupies $r + 5k$, so the difference between the cop and robber's positions is $5k$. If the robber plays $a$, then the cop will play $b$, and the difference between the cop and robber will decrease to $4k$. If the robber continues to play $a$, then the cop may continue to play $b$, decreasing the difference between the cop and robber's positions to $3k$, then $2k$, then $k$, and finally $0$. If the robber plays a different move $a'$, then the cop can also play $a'$ and maintain its difference with the robber.
• Figure 2: The figure shows a subgraph of an abelian Cayley digraph $\Gamma$. The generating set of $\Gamma$ contains six generating elements $a_1, \dots, a_6$ and six generating elements $b_1, \dots, b_6$, satisfying $a_1 - b_1 = \dots = a_6 - b_6$. Therefore, the group element $k = a_1 - b_1$ accounts for all six generating elements $a_1, \dots, a_6$, and thus if the robber occupies the vertex $r$, a cop at $r+k$ can guard all moves $a_1, \dots, a_6$.

Theorems & Definitions (50)

• Definition 2.1
• Theorem 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Lemma 2.7
• proof
• Lemma 2.8
• proof