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

Let \( G \) be a graph of order \( n \). The \emph{binding number} of \( G \) is defined as

\[
\text{bind}(G) := \min \left\{ \frac{|N_G(X)|}{|X|} \mid \emptyset \neq X \subseteq V(G) \text{ and } N_G(X) \neq V(G) \right\}.
\]

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.