A graph $G$ is $k$-vertex-critical if $χ(G)=k$ but $χ(G-v)<k$ for all $v\in V(G)$. In this paper we make progress on the open problem of the finiteness of $k$-vertex-critical $(P_4+\ell P_1)$-free graphs by showing that there are only finitely many $k$-vertex-critical graphs in the following subfamilies of $(P_4+\ell P_1)$-free graphs for all $k\ge 1$ and $\ell\ge 0$: $\bullet$ $(P_4+\ell P_1,\text{chair})$-free graphs, $\bullet$ $(P_4+\ell P_1,P_5,\text{bull})$-free graphs, and $\bullet$ $(P_4+\ell P_1,P_5,\text{cricket})$-free graphs. In fact, all but the first of these are special cases of our general result that there are only finitely many $k$-vertex-critical $(P_4+\ell P_1,B_{4}(m),B_{3}(m)^{+})$-free graphs for all $k\ge 1$ and $\ell,m\ge 0$. Here $B_{n}(m)$ is the graph obtained from a path of order $n$ by identifying one of its leaves with the centre vertex of $K_{1,m}$ and $B_{n}(m)^{+}$ is the graph obtained by identifying an edge of $K_3$ with the edge of $B_{n}(m)$ with endpoints of degrees $2$ and $m$, respectively. Our results imply the existence of simple polynomial-time certifying algorithms to decide the $k$-colourability of all graphs in these subfamilies for every fixed $k$. We also show that $χ(G)\le \ell+2$ for all $(P_4+\ell P_1,K_3)$-free graphs and all $\ell\ge 0$, improving the previously known upper bound of $2\ell+2$ that followed from Randerath and Schiermeyer's 2004 result on $(P_t,K_3)$-free graphs. More generally, we provide a $χ$-bound in $O(\ell^{ω-1})$ for $(P_4+\ell P_1)$-free graphs which improves the bound of $(2\ell+2)^{ω-1}$ which followed from Gravier, Hoàng and Maffray in 2003 for $P_{t}$-free graphs.

翻译：图 $G$ 称为 $k$-顶点临界图，若 $\chi(G)=k$ 但对所有 $v\in V(G)$ 有 $\chi(G-v)<k$。本文针对 $(P_4+\ell P_1)$-自由图中 $k$-顶点临界图有限性这一未解决问题取得进展，证明对于所有 $k\ge 1$ 及 $\ell\ge 0$，以下 $(P_4+\ell P_1)$-自由图子族中仅存在有限多个 $k$-顶点临界图：$\bullet$ $(P_4+\ell P_1,\text{chair})$-自由图，$\bullet$ $(P_4+\ell P_1,P_5,\text{bull})$-自由图，$\bullet$ $(P_4+\ell P_1,P_5,\text{cricket})$-自由图。事实上，除第一个子族外，其余子族均为我们的一般性结论的特例：对任意 $k\ge 1$ 及 $\ell,m\ge 0$，$(P_4+\ell P_1,B_{4}(m),B_{3}(m)^{+})$-自由图中仅存在有限多个 $k$-顶点临界图。其中 $B_{n}(m)$ 是将 $n$ 阶路的一个叶子顶点与 $K_{1,m}$ 的中心顶点重合所得到的图，$B_{n}(m)^{+}$ 是将 $K_3$ 的一条边与 $B_{n}(m)$ 中端点度数分别为 $2$ 和 $m$ 的边重合所得到的图。我们的结果意味着对于每个固定 $k$，存在简单的多项式时间验证算法来判断这些子族中所有图的 $k$-可着色性。我们还证明对所有 $(P_4+\ell P_1,K_3)$-自由图及所有 $\ell\ge 0$ 有 $\chi(G)\le \ell+2$，改进了此前由 Randerath 与 Schiermeyer 2004年关于 $(P_t,K_3)$-自由图结论导出的上界 $2\ell+2$。更一般地，我们给出 $(P_4+\ell P_1)$-自由图的 $O(\ell^{\omega-1})$ 色数上界，改进了 Gravier、Hoàng 与 Maffray 2003年关于 $P_t$-自由图结论导出的 $(2\ell+2)^{\omega-1}$ 上界。