Toughness and binding number bounds of star-like and path factor

Abstract

Let $\mathcal{L}$ be a set which consists of some connected graphs. Let $E$ be a spanning subgraph of graph $G$. It is called a $\mathcal{L}$-factor if every component of it is isomorphic to the element in $\mathcal{L}$. In this contribution, we give the lower bounds of four parameters ($t(G),$ $I(G), $ $I'(G),$ $\operatorname{bind}(G)$) of $G$, which force the graph $G$ admits a $(\{K_{1,i}:q\leq i\leq 2q-1\}\cup \{K_{2q+1}\})$-factor for $q\geq 2$ and a $\{P_2, P_{2q+1}\}$-factor for $q\geq 3$ respectively. The tightness of the bounds are given.