$k$-Critical Graphs in $P_5$-Free Graphs

TL;DR

This paper systematically classifies when the set of $k$-vertex-critical graphs is finite within subclasses of $P_5$-free graphs, providing a complete dichotomy for all four-vertex graphs $H$.

Contribution

It offers the first complete classification of the finiteness of $k$-vertex-critical graphs in $(P_5,H)$-free classes for all 4-vertex graphs $H$, using novel techniques.

Findings

Finiteness characterized for all $(P_5,H)$-free classes with $H$ on 4 vertices.

Proved finiteness for four new graphs $H$ using Ramsey-type and Dilworth's Theorem techniques.

Provides a complete dichotomy for the problem in the studied graph classes.

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 $H_2$. Let $P_t$ be the path on $t$ vertices. 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 $k$. The study of $k$-vertex-critical graphs for graph classes is an important topic in algorithmic graph theory because if the number of such graphs that are in a given hereditary graph class is finite, then there is a polynomial-time algorithm to decide if a graph in the class is $(k-1)$-colorable. In this paper, we initiate a systematic study of the finiteness of $k$-vertex-critical graphs in subclasses of $P_5$-free graphs. Our main result is a complete classification of the finiteness of $k$-vertex-critical graphs in the class of $(P_5,H)$-free graphs for all graphs $H$ on 4 vertices. To obtain the complete dichotomy, we prove the finiteness for four new graphs $H$ using various techniques -- such as Ramsey-type arguments and the dual of Dilworth's Theorem -- that may be of independent interest.