Cops and robber in graphs with bounded vertex cover number

TL;DR

The paper addresses Meyniel-type questions by bounding the cop number in terms of the vertex cover number $vc(G)$, proving for every connected graph $G$ that $cop(G) \le \frac{vc(G)}{2^{(1-o(1))\sqrt{\log vc(G)}}}$, a first sublinear bound in terms of such a structural parameter. The authors develop a general reduction framework that combines a deterministic geodesic-protection step with a ball-growth/probabilistic placement method, encapsulated in a main lemma bounding the protection cost of a vertex subset $X$ when components of $G-X$ have bounded diameter. They optimize the parameters and derive a corollary for vertex covers, while also discussing extensions to treedepth and posing open questions about cop-number bounds in other graph parameters. The approach blends classical protection of isometric subgraphs with probabilistic placement and Hall-type matching, enabling a sublinear bound that advances understanding of how graph structure constrains pursuit-evasion dynamics.

Abstract

Meyniel's conjecture states that $n$-vertex connected graphs have cop number $O(\sqrt{n})$. The current best known upper bound is $n/2^{(1-o(1))\sqrt{\log n}}$, proved independently by Lu and Peng (2011), and by Scott and Sudakov (2011). In this paper, we extend their result by showing that every connected graph with vertex cover number $k$ has cop number at most $k/2^{(1-o(1))\sqrt{\log k}}$. This is the first sublinear upper bound on the cop number in terms of the vertex cover number.

Theorems & Definitions (14)

• Theorem 1
• Theorem 2
• proof
• Lemma 4: Aigner84
• Lemma 5
• proof
• Theorem 6
• proof
• Lemma 7
• proof