On Quadrilaterals and $4$-path in Claw-free Graphs

Abstract

Let $G$ be a claw-free graph of order $4k$, where $k$ is a positive integer. In this paper, it is proved that if the degree sum $d(u)+d(v)$ is at least $4k–2$ for every pair of nonadjacent vertices $u,v\in V(G)$, then $G$ has a spanning subgraph consisting of $k–1$ quadrilaterals and a 4-path such that all of them are vertex-disjoint, unless $G$ is isomorphic to $K_{4k_1+2}\cup K_{4k_2+2}$, or $K_{4k_1+1}\cup K_{4k_2+3}$, where $k_1\ge 0,k_2\ge 0,k_1+k_2=k–1$. We further showed that the requirement about claw-free is indispensable and the degree condition is sharp.