On 2-factors in star-free graphs

Abstract

In this paper we give a sharp minimum degree condition for a $2$-connected star-free graph to have a $2$-factor containing specified edges. Let $G$ be a $2$-connected $K_{1,n}$-free graph with minimum degree $n+d$ and $I\subseteq E(G)$. If one of the followings holds, then $G$ has a $2$-factor which contains every edge in $I$: i) $n=3,d\ge 1,\left|I\right|\le 2$ and $|V(G)|\ge 8$ if $\left|I\right|=2$; ii) $n=4,d\ge 1,\left|I\right|\le 2$ and $|V(G)|\ge 11$ if $\left|I\right|=2$; iii) $n\ge 5,d\ge 1$ and $\left|I\right|\le 1$; iv) $n\ge 5,d\ge \lfloor (\sqrt{4n-3}+1)/2\rfloor$ and $\left|I\right|\le 2$.