Vertex-critical $(P_3+\ell P_1)$-free and vertex-critical (gem, co-gem)-free graphs

A graph $G$ is $k$-vertex-critical if $\chi(G)=k$ but $\chi(G-v)<k$ for all $v\in V(G)$ where $\chi(G)$ denotes the chromatic number of $G$. We show that there are only finitely many $k$-critical $(P3+\ell P1)$-free graphs for all $k$ and all $\ell$. Together with previous results, the only graphs $H$ for which it is unknown if there are an infinite number of $k$-vertex-critical $H$-free graphs is $H=(P4+\ell P1)$ for all $\ell\ge 1$. We consider a restriction on the smallest open case, and show that there are only finitely many $k$-vertex-critical (gem, co-gem)-free graphs for all $k$, where gem$=\overline{P4+P1}$. To do this, we show the stronger result that every vertex-critical (gem, co-gem)-free graph is either complete or a clique expansion of $C_5$. This characterization allows us to give the complete list of all $k$-vertex-critical (gem, co-gem)-free graphs for all $k\le 16$