论文标题
$ k_ {1，r} $的组件因子 - 免费图形
Component factors in $K_{1,r}$-free graphs
论文作者
论文摘要
据说图形为$ k_ {1，r} $ - 如果不包含诱导子图的同构为$ k_ {1，r} $，则免费。 $ \ Mathcal {f} $ - 因子是一个子图$ h $，因此$ h $的每个连接组件对$ \ nathcal {f} $中的某些图都是同构的。特别是，$ h $称为$ \ {p_2，p_3 \} $ - $ g $的因子，如果$ \ mathcal {f} = \ {p_2，p_3 \} $; $ h $称为$ \ MATHCAL {s} _n $ - $ g $如果$ \ Mathcal {f} = \ {k_ {1,1}，k_ {1,2}，k_ {1,2}，k_ {1,3}，...图$ g $的跨度子图称为$ \ mathcal {p} _ {\ geq k} $ - $ g $的因子，如果其每个组件对订单途径均等至少$ k $，其中$ k \ geq2 $。图形$ g $称为$ \ MATHCAL {F} $ - 因子覆盖图，如果有$ \ Mathcal {f} $ - $ g $的因子，包括$ e $，包括E（g）$中的任何$ e \。在本文中，我们给出了$ k_ {1，r} $的最低度条件 - 免费图形具有$ \ nathcal {s} _n $ - factor和$ \ nathcal {p} _ {\ geq 3} $ - 分别为factor。此外，我们获得了$ k_ {1，r} $的足够条件 - 免费图形为$ \ MATHCAL {p} _ {\ geq 2} $ - factor，$ \ Mathcal {p} _ {\ geq 3} $ - factor或$ \ factor或$ \ {p_2，p_2，p_3 \ \ \ \ \ \ f is offers oprage offers offers office。此外，示例表明我们的结果很清晰。
A graph is said to be $K_{1,r}$-free if it does not contain an induced subgraph isomorphic to $K_{1,r}$. An $\mathcal{F}$-factor is a spanning subgraph $H$ such that each connected component of $H$ is isomorphic to some graph in $\mathcal{F}$. In particular, $H$ is called an $\{P_2,P_3\}$-factor of $G$ if $\mathcal{F}=\{P_2,P_3\}$; $H$ is called an $\mathcal{S}_n$-factor of $G$ if $\mathcal{F}=\{K_{1,1},K_{1,2},K_{1,3},...,K_{1,n}\}$, where $n\geq2$. A spanning subgraph of a graph $G$ is called a $\mathcal{P}_{\geq k}$-factor of $G$ if its each component is isomorphic to a path of order at least $k$, where $k\geq2$. A graph $G$ is called a $\mathcal{F}$-factor covered graph if there is a $\mathcal{F}$-factor of $G$ including $e$ for any $e\in E(G)$. In this paper, we give a minimum degree condition for a $K_{1,r}$-free graph to have an $\mathcal{S}_n$-factor and a $\mathcal{P}_{\geq 3}$-factor, respectively. Further, we obtain sufficient conditions for $K_{1,r}$-free graphs to be $\mathcal{P}_{\geq 2}$-factor, $\mathcal{P}_{\geq 3}$-factor or $\{P_2,P_3\}$-factor covered graphs. In addition, examples show that our results are sharp.
