Some Results on $k$-Critical $P_5$-Free Graphs

TL;DR

This paper proves finiteness results for $k$-vertex-critical graphs within certain hereditary classes defined by forbidden induced subgraphs, and characterizes these graphs for specific values of $k$ using computational methods.

Contribution

It establishes the finiteness of $k$-vertex-critical ($P_5$,gem)-free and ($P_5$,overline{P_3+P_2})-free graphs for all $k$, and characterizes such graphs for $k=4,5$.

Findings

Finiteness of $k$-vertex-critical graphs in specified classes.

Complete characterization for $k=4,5$ in these classes.

Structural insights for ($P_5$,gem)-free graphs.

Abstract

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 prove that for every fixed integer $k\ge 1$, there are only finitely many $k$-vertex-critical ($P_5$,gem)-free graphs and $(P_5,\overline{P_3+P_2})$-free graphs. To prove the results we use a known structure theorem for ($P_5$,gem)-free graphs combined with properties of $k$-vertex-critical graphs. Moreover, we characterize all $k$-vertex-critical ($P_5$,gem)-free graphs and $(P_5,\overline{P_3+P_2})$-free graphs for $k \in \{4,5\}$ using a computer generation algorithm.