Antifactors of regular bipartite graphs

Abstract

Let G = (X, Y ) be a bipartite graph. Does G contain a factor F such that d F (v) = 1 for all v ∈ X and d F (v) = 1 for all v ∈ Y ? Lovász and Plummer (Matching Theory, Ann. Discrete Math., 29 North-Holland, Amsterdam, 1986.) asked whether this problem is polynomially solvable and an affirmative answer was given by Cornuéjols (General factors of graphs, J. Combin. Theory Ser. B, 45 (1988), 185-198). Let k ≥ 3 be an integer. Liu and Yu asked whether every k-regular bipartite graph G contains such a factor F . In this paper, we solve the question of Liu and Yu in the affirmative.