Some results on star-factor deleted graphs

Abstract

Let $G$ be a graph and let $k\geq2$ be an integer. A $\{K_{1,j}:1\leq j\leq k\}$-factor of $G$ is a spanning subgraph of $G$, in which every component is isomorphic to a member in $\{K_{1,j}:1\leq j\leq k\}$. A graph $G$ is called a $\{K_{1,j}:1\leq j\leq k\}$-factor deleted graph if for any $e\in E(G)$, $G$ has a $\{K_{1,j}:1\leq j\leq k\}$-factor excluding $e$. In this article, we first give a characterization of $\{K_{1,j}:1\leq j\leq k\}$-factor deleted graph. Then we show a lower bound on the binding number (resp. the size) of $G$ to ensure that $G$ is a $\{K_{1,j}:1\leq j\leq k\}$-factor deleted graph. Finally, we construct two extremal graphs to claim that the bounds derived in this article are sharp.