论文标题
顶点临界$（p_3+\ ell p_1）$ - free and vertex-clitical（gem，coem） - 免费图形
Vertex-critical $(P_3+\ell P_1)$-free and vertex-critical (gem, co-gem)-free graphs
论文作者
论文摘要
如果$χ（g）= k $，则图$ g $是$ k $ - vertex--关键的，但$χ（g-v）<k $的所有$ v \ in v（g）$中，其中$χ（g）$表示$ g $的色数。我们表明，只有许多$ k $ -Critical $（p_3+\ ell p_1）$ - 所有$ k $和所有$ \ ell $的免费图。与先前的结果一起，唯一的图形$ h $，如果有无限的$ k $ -vertex-Critical $ h $ -free Graphs为$ h =（p_4+\ ell p_1）$。我们考虑对最小的开放式案例的限制，并表明只有许多$ k $ - vertex-Critical（GEM，Co-Gem） - 所有$ k $的free Graphs，其中GEM $ = \ overline {p_4+p_1} $。为此，我们表明了每个关键顶点（GEM，Co-Gem）的较强结果是完整的，要么是$ C_5 $的集团扩展。这种表征使我们能够提供所有$ k $ -vertex-Contrical-timcitical（GEM，Co-Gem）的完整列表 - 所有$ k \ le 16 $的免费图
A graph $G$ is $k$-vertex-critical if $χ(G)=k$ but $χ(G-v)<k$ for all $v\in V(G)$ where $χ(G)$ denotes the chromatic number of $G$. We show that there are only finitely many $k$-critical $(P_3+\ell P_1)$-free graphs for all $k$ and all $\ell$. 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 $H=(P_4+\ell P_1)$ for all $\ell\ge 1$. 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 $k\le 16$
