Path Cover in $K_{1,4}$-Free Graphs

DOI：10.3770/j.issn:2095-2651.2019.03.011

 作者 单位 刘明达 辽宁工业大学理学院, 辽宁 锦州 121001 陈晓东 辽宁工业大学理学院, 辽宁 锦州 121001 李明楚 大连理工大学软件学院, 辽宁 大连 116024

图$G$的路覆盖为该图的一个由点不交的路而构成的$G$的生成子图; 路覆盖数, 记为$p(G)$, 表示$G$的所有路覆盖中含有路的条数最少的路覆盖所含有的路的条数.本篇论文证明了若图$G$为一个阶数为$n$,无$K_{1,4}$子图的图,且$\delta_{k+1}(G)\ge n-k$,则图$G$的路覆盖数至多为$k$.

For a graph $G,$ a path cover is a set of vertex disjoint paths covering all the vertices of $G$, and a path cover number of $G,$ denoted by $p(G),$ is the minimum number of paths in a path cover among all the path covers of $G.$ In this paper, we prove that if $G$ is a $K_{1,4}$-free graph of order $n$ and $\sigma_{k+1}(G)\geq {n-k}$, then $p(G)\leq k$, where $\sigma_{k+1}(G)=\min\{\sum_{v\in S}{\rm d}(v):S$ is an independent set of $G$ with $|S|=k+1\}$.