4-cop-win graphs have at least 19 vertices

TL;DR

This paper proves that all graphs with 18 or fewer vertices can be won by three cops in the game, and it identifies specific 3-cop-win graphs on 11 vertices, advancing understanding of cop-win graph properties.

Contribution

It establishes a new upper bound on the size of graphs requiring four cops, answering longstanding questions and identifying all 3-cop-win graphs on 11 vertices.

Findings

All graphs with 18 or fewer vertices are 3-cop-win.

Complete enumeration of 3-cop-win graphs on 11 vertices.

Progress on minimal order of 3-cop-win planar graphs.

Abstract

We show that the cop number of any graph on 18 or fewer vertices is at most 3. This answers a question posed by Andreae in 1986, as well as more recently by Baird et al. We also find all 3-cop-win graphs on 11 vertices, narrow down the possible 4-cop-win graphs on 19 vertices and make some progress on finding the minimum order of 3-cop-win planar graphs.