Binding Number, Minimum Degree for Connected $(g,f+I)$-factors in Graphs

Abstract

Let $G$ be a graph of order $n$. The $bindingnumber$ of $G$ is defined as

$$\text{bind}(G):=min\{\frac{|N_G(X)|}{|X|}\mid \mathrm{\varnothing }\ne X\subseteq V(G)\text{ and }{N}_G(X)\ne V(G)\}.$$

A $(g,f)$-factor is called a connected $(g,f)$-factor if it is connected. A $(g,f)$-factor $F$ is called a Hamilton $(g,f)$-factor if $F$ contains a Hamilton cycle. In this paper, several sufficient conditions related to binding number and minimum degree for graphs to have connected $(g,f+1)$-factors or Hamilton $(g,f)$-factors are given.