The Petersen graph is the smallest 3-cop-win graph

TL;DR

This paper provides a rigorous mathematical proof that the Petersen graph is the smallest connected graph requiring three cops to catch a robber, confirming previous computational findings and exploring properties of graphs with higher cop numbers.

Contribution

The paper offers a self-contained proof that the Petersen graph uniquely has the smallest order with cop number three, and characterizes graphs with higher cop numbers and maximum degree.

Findings

Petersen graph is the smallest 3-cop-win graph

Characterizations of graphs with cop number greater than 2

Insights into graphs with high maximum degree

Abstract

In the game of \emph{cops and robbers} on a graph $G = (V,E)$, $k$ cops try to catch a robber. On the cop turn, each cop may move to a neighboring vertex or remain in place. On the robber's turn, he moves similarly. The cops win if there is some time at which a cop is at the same vertex as the robber. Otherwise, the robber wins. The minimum number of cops required to catch the robber is called the \emph{cop number} of $G$, and is denoted $c(G)$. Let $m_k$ be the minimum order of a connected graph satisfying $c(G) \geq k$. Recently, Baird and Bonato determined via computer search that $m_3=10$ and that this value is attained uniquely by the Petersen graph. Herein, we give a self-contained mathematical proof of this result. Along the way, we give some characterizations of graphs with $c(G) >2$ and very high maximum degree.