Vertex-critical $(P_3+\mathrm{\ell }{P}_1)$-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 $(P_3+\mathrm{\ell }{P}_1)$-free graphs for all $k$ and all $\mathrm{\ell }$. 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 $H=(P_4+\mathrm{\ell }{P}_1)$ for all $\mathrm{\ell }\ge 1$. 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$=\overline{P_4+P_1}$. 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 $k\le 16$