Dichotomizing $k$-vertex-critical $H$-free graphs for $H$ of order four

TL;DR

This paper characterizes the finiteness of $k$-vertex-critical $H$-free graphs for graphs $H$ of order four and introduces polynomial-time algorithms for $k$-colorability in specific graph classes.

Contribution

It proves finiteness results for certain $k$-vertex-critical graphs and characterizes all such graphs for fixed $k \, \geq 5$, also providing new algorithms.

Findings

Finite number of $k$-vertex-critical $(P_2+\ell P_1)$-free graphs.

Bound of $2k-1$ vertices for $k$-vertex-critical $(P_3+P_1)$-free graphs.

Characterization of $H$-free graphs with finite $k$-vertex-critical graphs for fixed $k \geq 5$.

Abstract

For $k \geq 3$, we prove (i) there is a finite number of $k$-vertex-critical $(P_2+\ell P_1)$-free graphs and (ii) $k$-vertex-critical $(P_3+P_1)$-free graphs have at most $2k-1$ vertices. Together with previous research, these results imply the following characterization where $H$ is a graph of order four: There is a finite number of $k$-vertex-critical $H$-free graphs for fixed $k \geq 5$ if and only if $H$ is one of $\overline{K_4}, P_4, P_2 + 2P_1$, or $P_3 + P_1$. Our results imply the existence of new polynomial-time certifying algorithms for deciding the $k$-colorability of $(P_2+\ell P_1)$-free graphs for fixed $k$.