AI 中文总结
本文证明当H为二部图且k≥5时，不含P₅和H的k-顶点临界图数量仅有限个当且仅当H不含2P₂，还得出相关图类的有限性/无穷性结论，提出了新的多项式时间可验证着色算法。
AI 中文摘要
若图G满足χ(G)=k，且对G的每个诱导子图H都有χ(H)<k，则称G为k-顶点临界图；若图G不含任何Hᵢ（i∈{1,2,…,m}）作为诱导子图，则称G为不含(H₁,H₂,…,Hₘ)的图。本文证明了如下二分性结论：对于二部图H及任意固定整数k≥5，不含P₅和H的k-顶点临界图仅有限个当且仅当H不含2P₂。基于此，本文提出了一个问题：对于满足χ(H)≥3的图H，哪些类别的H会使得对所有k≥5，都存在无穷多个不含P₅和H的k-顶点临界图？针对该问题，本文证明：对所有k,s,t≥1，不含P₅和K_{s,t}+e（K_{s,t}为完全二部图，K_{s,t}+e为其添加一条边后的图）的k-顶点临界图仅有限个；另一方面，对所有k≥5，不含P₅和网图（net）、余网图（co-net）、补5环（$\boldsymbol{\bar{C}_5}$）、补6环（$\boldsymbol{\bar{C}_6}$）、…、补(k-1)环（$\boldsymbol{\bar{C}_{k-1}}$）的k-顶点临界图有无穷多个。本文还证明：对所有ℓ,n≥0，不含P₄+ℓ个孤立点（P₄+ℓP₁）和补完全二部图$\boldsymbol{\bar{L(K_{2,n})}}$的k-顶点临界图仅有限个，这为不含P₄+ℓP₁的图中满足该性质的最大已知子族提供了结果。结合已有研究，本文的结论意味着存在新的多项式时间可验证算法，可用于判定固定k≥5时，大量不含P₅和不含P₄+ℓP₁的图子族的k-可着色性。本文的证明技巧运用了Chudnovsky、Kim、Oum和Seymour（2016）关于素图的重要定理，该定理有望在其他遗传图类的k-顶点临界图数量界定问题中得到进一步应用。
英文摘要
A graph $G$ is $k$-vertex-critical if $χ(G)=k$, but $χ(H)<k$ for every induced subgraph $H$ of $G$. A graph $G$ is $(H_1,H_2,\dots,H_m)$-free if does not contain $H_i$ as an induced subgraph for any $i\in\{1,2,\dots,m\}$.We provide the following dichotomy that for bipartite graphs $H$ and any fixed integer $k\ge 5$ , there are only finitely many $k$-vertex-critical $(P_5,H)$-free graphs if and only if $H$ is $2P_2$-free. This leads us to pose the problem about determining for which graphs $H$ with $χ(H)\ge 3$ there are infinitely many $k$-vertex-critical $(P_5,H)$-free graphs for all $k\ge 5$. Toward this problem, we show that there only finitely many $k$-vertex-critical $(P_5, K_{s,t}+e)$-free graphs for all $k,s,t\ge 1$, where $K_{s,t}+e$ is a complete bipartite graph plus a single edge. On the other hand, we show that there are infinitely many $k$-vertex-critical $(P_5,\operatorname{net},\operatorname{co-net},\overline{C_5},\overline{C_6},\dots\overline{C_{k-1}})$-free graphs for all $k\ge 5$. We also show that there are only finitely many $k$-vertex-critical $(P_4+\ell P_1,\overline{L(K_{2,n})})$-free graphs for all $\ell,n\ge 0$, providing the largest known subfamily of $(P_4+\ell P_1)$-free graphs to satisfy this property. Our results, together with known results, imply the existence of new polynomial-time certifying algorithms to determine the $k$-colourability of many subfamilies of $P_5$-free and $(P_4+\ell P_1)$-free graphs for fixed $k\ge 5$. Our proof techniques apply a powerful theorem of Chudnovsky, Kim, Oum, and Seymour (2016) on prime graphs that we expect to be of interest and have further applications to bounding the number of $k$-vertex-critical graphs in other hereditary families of graphs.
