Disjoint stars and forbidden subgraphs

Abstract

Let $r,k$ be integers with $r\ge 3, k\ge 2$. We prove that if $G$ is a $K_{1,r}$-free graph of order at least $(k-1)(2r-1)+1$ with $\delta(G)\ge 2$, then $G$ contains $k$ vertex-disjoint copies of $K_{1,2}$. This result is motivated by the problem of characterizing a forbidden subgraph $H$ which satisfies the statement "every $H$-free graph of sufficiently large order with minimum degree at least $t$ contains $k$ vertex-disjoint copies of a star $K_{1,t}$." In this paper, we also give the answer to this problem.