Minimum Path Cover in Quasi-claw-Free Graphs

Abstract

A path cover of a graph G is a spanning subgraph of G consisting of vertex disjoint paths; a minimum path cover of G is a path cover of G consisting of minimum number of paths; a path cover number of G, denoted by p(G), is the number of paths in a minimum path cover of G, i.e., $$p(G)=\min \{|{\mathcal {P}}|:$$$${\mathcal {P}}$$ is a path cover of $$G\}.$$ A graph G is quasi-claw-free if for any two vertices x, y with $$d(x,y)=2$$ in G, there exists a vertex u with $$u\in N(x)\cap N(y)$$ and $$N(u)\subseteq N[x]\cup N[y].$$ In this paper, we prove that for any quasi-claw-free graph G of order n, if $$\sigma _{k+1}(G)\ge {n-k}$$ for a positive integer k, then $$p(G)\le k,$$ where $$\sigma _{k+1}(G)$$ is the minimum degree sum of an independent set with $$k+1$$ vertices in G. Our result is a generalization of Ore’s sufficient conditions of hamiltonicity for quasi-claw-free graphs.