Some sufficient conditions for a graph to be strong star factor

Abstract

Let $G$ be a graph and let $r⩾2$ be an integer. We call a spanning subgraph $H$ of a graph $G$ a strong $\{K_{1,1},K_{1,2},\dots ,K_{1,r}\}$-factor if every component of $H$ is isomorphic to an element of $\{K_{1,1},K_{1,2},\dots ,K_{1,r}\}$ and is an induced subgraph of $G$. A $\{K_{1,1},K_{1,2},\dots ,K_{1,r}\}$-factor of $G$ is a spanning subgraph of $G$, in which each component is isomorphic to a member in $\{K_{1,1},K_{1,2},\dots ,K_{1,r}\}$. If $G$ contains a strong $\{K_{1,1},K_{1,2},\dots ,K_{1,r}\}$-factor, then $G$ contains a $\{K_{1,1},K_{1,2},\dots ,K_{1,r}\}$-factor. In this paper, through the typical spectral techniques, we establish respectively a lower bound on the size and a lower bound on the spectral radius to ensure whether or not a graph admits a strong $\{K_{1,1},K_{1,2},\dots ,K_{1,r}\}$-factor (resp. a $\{K_{1,1},K_{1,2},\dots ,K_{1,r}\}$-factor).