A graph $G$ is $k$-vertex-critical if $\chi(G)=k$ but $\chi(G-v)<k$ for all $v\in V(G)$ and $(G,H)$-free if it contains no induced subgraph isomorphic to $G$ or $H$. We show that there are only finitely many $k$-vertex-critical (co-gem, $H$)-free graphs for all $k$ when $H$ is any graph of order $4$ by showing finiteness in the three remaining open cases, those are the cases when $H$ is $2P_2$, $K_3+P_1$, and $K_4$. For the first two cases we actually prove the stronger results: $\bullet$ There are only finitely many $k$-vertex-critical (co-gem, paw$+P_1$)-free graphs for all $k$ and that only finitely many $k$-vertex-critical (co-gem, paw$+P_1$)-free graphs for all $k\ge 1$. $\bullet$ There are only finitely many $k$-vertex-critical (co-gem, $P_5$, $P_3+cP_2$)-free graphs for all $k\ge 1$ and $c\ge 0$. To prove the latter result, we employ a novel application of Sperner's Theorem on the number of antichains in a partially ordered set. Our result for $K_4$ uses exhaustive computer search and is proved by showing the stronger result that every $(\text{co-gem, }K_4)$-free graph is $4$-colourable. Our results imply the existence of simple polynomial-time certifying algorithms to decide the $k$-colourability of (co-gem, $H$)-free graphs for all $k$ and all $H$ of order $4$ by searching the vertex-critical graphs as induced subgraphs.

翻译：若图$G$满足$\chi(G)=k$但对所有$v\in V(G)$有$\chi(G-v)<k$，则称其为$k$-顶点临界图；若图不包含任何与$G$或$H$同构的诱导子图，则称其为$(G,H)$-free图。我们证明当$H$是任意4阶图时，对所有$k$值而言，$k$-顶点临界（co-gem, $H$）-free图的数量总是有限的。这一结论通过解决最后三个未决情形得以完成，即$H$为$2P_2$、$K_3+P_1$和$K_4$的情况。针对前两种情形，我们实际上证明了更强的结论：$\bullet$ 对所有$k$值，$k$-顶点临界（co-gem, paw$+P_1$）-free图的数量有限；且对所有$k\ge 1$，$k$-顶点临界（co-gem, paw$+P_1$）-free图的数量有限。$\bullet$ 对所有$k\ge 1$和$c\ge 0$，$k$-顶点临界（co-gem, $P_5$, $P_3+cP_2$）-free图的数量有限。为证明后一结论，我们创新性地应用了Sperner定理中关于偏序集反链数量的理论。针对$K_4$情形的证明采用穷举计算机搜索，并通过证明更强的结论实现：每个（co-gem, $K_4$）-free图都是$4$-可着色的。我们的研究结果意味着，对于所有$k$值和所有4阶图$H$，存在简单的多项式时间可验证算法，通过搜索顶点临界图作为诱导子图，能够判定（co-gem, $H$）-free图的$k$-可着色性。