Generalizations of Marriage Theorem for Degree Factors

Abstract

Let G be a bipartite graph with bipartition (A, B). We give new criteria for a bipartite graph to have an f -factor, a (g, f)-factor and other factors together with some applications of these criteria. These criteria can be considered as direct generalizations of Hall’s marriage theorem. Among some results, we prove that for a function $h: A\cup B \rightarrow \{0,1,2, \ldots \}$, G has a factor F such that $\deg _F(x)=h(x)$ for $x\in A$ and $\deg _H(y) \le h(y)$ for $y\in B$ if and only if $h(X) \le \sum _{x\in N_G(X)}\min \{h(x), e_G(x,X)\}$ for all $X\subseteq A$.