A Note on Path Factors in Claw-Free Graphs

Abstract

A graph $G$ is called $claw-free$ if $G$ has no induced subgraph isomorphic to $K_{1,3}$. Ando et al. obtained the result: a claw-free graph $G$ with minimum degree at least $d$ has a path-factor such that the order of each path is at least $d+1$; in particular $G$ has a $\{P_3,P_4,P_5\}$-factor whenever $d\ge 2$. Kawarabayashi et al. proved that every $2$-connected cubic graph has a $\{P_3,P_4\}$-factor. In this article, we show that if $G$ is a connected claw-free graph with at least $6$ vertices and minimum degree at least $2$, then $G$ has a $\{P_3,P_4\}$-factor. As an immediate consequence, it follows that every claw-free cubic graph (not necessarily connected) has a $\{P_3,P_4\}$-factor.