A note onf-factors in directed and undirected multigraphs

Abstract

We prove that a graphG admits a 2-factorF whenever there is an edgeuv such that bothG ? u andG ? v have a 2-factor; moreover we can takeF to be contained in the union of these 2-factors (and the edgeuv) and to contain their intersection. These results are generalized to certainf-factors and analogous results are obtained for bipartite and for directed graphs.