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

Abstract:

      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 $|S|=k+1\}$.