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 $$h: A\cup B \rightarrow \{0,1,2, \ldots \}$$h:AźBź{0,1,2,ź}, G has a factor F such that $$\deg _F(x)=h(x)$$degF(x)=h(x) for $$x\in A$$xźA and $$\deg _H(y) \le h(y)$$degH(y)≤h(y) for $$y\in B$$yźB if and only if $$h(X) \le \sum _{x\in N_G(X)}\min \{h(x), e_G(x,X)\}$$h(X)≤źxźNG(X)min{h(x),eG(x,X)} for all $$X\subseteq A$$X⊆A.