Vertex-Critical $(P_5, chair)$-Free Graphs

TL;DR

This paper proves that there are finitely many 5-vertex-critical graphs that are free of both $P_5$ and chair subgraphs, contributing to the understanding of graph coloring constraints in restricted graph classes.

Contribution

It establishes the finiteness of 5-vertex-critical $(P_5,chair)$-free graphs, a new result in the study of graph colorings and forbidden subgraph classes.

Findings

Finiteness of 5-vertex-critical $(P_5,chair)$-free graphs proved

Advances understanding of graph coloring in restricted classes

Provides groundwork for classification of critical graphs

Abstract

Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. A $P_t$ is the path on $t$ vertices. A chair is a $P_4$ with an additional vertex adjacent to one of the middle vertices of the $P_4$. A graph $G$ is $k$-vertex-critical if $G$ has chromatic number $k$ but every proper induced subgraph of $G$ has chromatic number less than $k$. In this paper, we prove that there are finitely many 5-vertex-critical $(P_5,chair)$-free graphs.