Obstructions for three-coloring graphs without induced paths on six vertices

TL;DR

This paper classifies all 4-critical graphs that are free of induced paths on six vertices, providing a complete list of such graphs and exploring the conditions for their existence.

Contribution

The authors completely enumerate 24 4-critical $P_6$-free graphs and establish conditions for the infinitude of 4-critical $H$-free graphs when $H$ is connected and not a subgraph of $P_6$.

Findings

Identified exactly 24 4-critical $P_6$-free graphs.

Provided a complete list of these graphs.

Showed infinite families of 4-critical $H$-free graphs when $H$ is connected and not a subgraph of $P_6$.

Abstract

We prove that there are 24 4-critical $P_6$-free graphs, and give the complete list. We remark that, if $H$ is connected and not a subgraph of $P_6$, there are infinitely many 4-critical $H$-free graphs. Our result answers questions of Golovach et al. and Seymour.