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4-cop-win graphs have at least 19 vertices

Jérémie Turcotte, Samuel Yvon

TL;DR

This work resolves a long-standing question about the minimum size of a 4-cop-win graph by proving $M_4=19$, while also showing that all graphs on at most 18 vertices are 3-cop-win. The authors combine deep structural analysis around the Petersen and cornered Petersen graphs with extensive computer-assisted enumeration and a novel merging algorithm to generate and test candidate 4-cop-win graphs. They classify all 3-cop-win graphs on small orders, establish edge-count and degree-based barriers that exclude most high-degree configurations, and isolate the Robertson graph as the unique 4-cop-win graph on 19 vertices under broad conditions. The results, aided by open-source code and a systematic phase-based approach, advance our understanding of Meyniel-type bounds and demonstrate a concrete pathway for tackling higher-order cop-win problems.

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.

4-cop-win graphs have at least 19 vertices

TL;DR

This work resolves a long-standing question about the minimum size of a 4-cop-win graph by proving , while also showing that all graphs on at most 18 vertices are 3-cop-win. The authors combine deep structural analysis around the Petersen and cornered Petersen graphs with extensive computer-assisted enumeration and a novel merging algorithm to generate and test candidate 4-cop-win graphs. They classify all 3-cop-win graphs on small orders, establish edge-count and degree-based barriers that exclude most high-degree configurations, and isolate the Robertson graph as the unique 4-cop-win graph on 19 vertices under broad conditions. The results, aided by open-source code and a systematic phase-based approach, advance our understanding of Meyniel-type bounds and demonstrate a concrete pathway for tackling higher-order cop-win problems.

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.

Paper Structure

This paper contains 18 sections, 24 theorems, 10 equations, 7 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Notation and previous results
  3. Computational results for small 3-cop-win graphs
  4. Graphs with high maximum degree
  5. Graphs with maximum degree 3
  6. Remaining cases
  7. Presentation of the Merging Algorithm
  8. Quick Overview
  9. Input of the Algorithm
  10. Output of the Algorithm
  11. Required definitions
  12. Phase 1 of the Algorithm
  13. Phase 2 of the Merging Algorithm
  14. Comments on the Merging Algorithm
  15. Validity of the Merging Algorithm
  16. ...and 3 more sections

Key Result

Theorem 2.1

andreae_pursuit_1986baird_minimum_2014 Let $G$ be a connected graph. In particular, $M_3=10$.

Figures (7)

  • Figure 1: Some small $(d,5)$-cage graphs
  • Figure 2: The cornered Petersen graphs
  • Figure 3: Typical application of the chasing strategy on the Petersen graph.
  • Figure 4: Example situation during the proof of Lemma \ref{['mindeg4']}. Unused or unknown vertices and edges are omitted.
  • Figure 5: Example situation during the proof of Lemma \ref{['neighbourhoodproperty']}. Unused or unknown vertices and edges are omitted.
  • ...and 2 more figures

Theorems & Definitions (50)

  • Theorem 2.1
  • Lemma 2.2
  • Definition 2.3
  • Theorem 2.4
  • Corollary 2.5
  • proof
  • Corollary 2.6
  • proof
  • Definition 2.7
  • Corollary 2.8
  • ...and 40 more