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arXiv 2408.05027math.COcs.DM
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无co-gem图中的顶点临界图

Vertex-critical graphs in co-gem-free graphs

Iain Beaton, Ben Cameron

备注：
Replaced Theorem 3.1 with a stronger statement in version 2
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AI总结：

该研究证明了当H为任意4阶图时，无(co-gem, H)图中k-顶点临界图的数量有限，还得到了若干更强结论，并由此推出判定这类图k可着色性的多项式时间认证算法的存在性。

摘要：

若图$G$满足$χ(G)=k$，但对所有$v∈V(G)$都有$χ(G-v)<k$，则称$G$为$k$-顶点临界图；若图不含同构于$G$或$H$的诱导子图，则称其为无$(G,H)$图。我们通过证明三个剩余未决情形的有限性，得出当$H$为任意4阶图时，对所有$k$，仅存在有限个$k$-顶点临界无(co-gem, $H$)图；这三个情形分别是$H$为$2P_2$、$K_3+P_1$和$K_4$的情况。对于前两种情形，我们实际上证明了更强的结论：一是对所有$k$，仅存在有限个$k$-顶点临界无(co-gem, paw$+P_1$)图，且对所有$k≥1$，$k$-顶点临界无(co-gem, paw$+P_1$)图的数量有限；二是对所有$k≥1$和$c≥0$，仅存在有限个$k$-顶点临界无(co-gem, $P_5$, $P_3+cP_2$)图。为证明后一个结论，我们创新性地应用了关于偏序集中反链数量的Sperner定理（斯佩纳定理）。针对$K_4$的结果采用了穷举计算机搜索，通过证明每个无(co-gem, $K_4$)图都是4可着色的这一更强结论得证。我们的结果意味着，通过搜索作为诱导子图的顶点临界图，存在简单的多项式时间认证算法，可判定所有$k$和所有4阶$H$对应的无(co-gem, $H$)图的$k$可着色性。

英文摘要：

A graph $G$ is $k$-vertex-critical if $χ(G)=k$ but $χ(G-v)<k$ 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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