Vertex-critical co-gem-free graphs

Abstract

A graph $G$ is $k$-$colorable$ if $V(G)$ can be partitioned into at most $k$ stable sets. A graph $G$ is $k$-$chromatic$ if $k$ is the smallest integer for which $G$ is $k$-colorable. In general, for a fixed $k\ge 3$, determining whether an arbitrary graph $G$ is $k$-colorable is NP-complete. Consequently, $k$-coloring algorithms for restricted graph classes, such as $\mathcal{H}$-free graphs, have been widely studied over the past few decades. A graph $G$ is $k$-$vertex$-$critical$ if $G$ is $k$-chromatic and every proper induced subgraph of $G$ is ($k$-1)-colorable. Given a graph $G$, most of the certifying $k$-coloring algorithms in the literature either output a $k$-coloring of $G$ or a ($k$+1)-vertex-critical induced subgraph of $G$, thus, proving that $G$ is not $k$-colorable. As a result, $k$-vertex-critical graphs have gathered considerable attention in the recent years. Beaton and Cameron [Vertex-critical graphs in co-gem-free graphs, Theoretical Computer Science 1042 (2025) 115234] asked for which graphs $H$ of order five are there finitely many $k$-vertex-critical (co-gem, $H$)-free graphs for all $k$? In this paper we explore the structure of (co-gem, house)-free graphs and (co-gem, dart)-free graphs, and prove that, for each $k\ge 1$, there are finitely many $k$-vertex-critical (co-gem, $H$)-free graphs, when $H$ is in $\{$house, dart$\}$.