On $2$-Factors in $r$-Connected $\{K_{1,k},P_4\}$-Free Graphs

Abstract

In [3], Faudree et~al.\ considered the proposition ``Every $\{X,Y\}$-free graph of sufficiently large order has a $2$-factor,'' and they determined those pairs $\{X,Y\}$ which make this proposition true. Their result says that one of them is $\{X,Y\}=\{K_{1,4},P_4\}$. In this paper, we investigate the existence of $2$-factors in $r$-connected $\{K_{1,k},P_4\}$-free graphs. We prove that if $r\ge 1$ and $k\ge 2$, and if $G$ is an $r$-connected $\{K_{1,k},P_4\}$-free graph with minimum degree at least $k-1$, then $G$ has a $2$-factor with at most $\max\{k-r,1\}$ components unless $(k-1) K_2 + (k-2) K_1 \subseteq G \subseteq (k-1) K_2 + K_{k-2}$. The bound on the minimum degree is best possible.