Antifactors of regular bipartite graphs

Abstract

Let $G=(X,Y;E)$ be a bipartite graph, where $X$ and $Y$ are color classes and $E$ is the set of edges of $G$. Lovász and Plummer \cite{LoPl86} asked whether one can decide in polynomial time that a given bipartite graph $G=(X,Y;E)$ admits a 1-anti-factor, that is subset $F$ of $E$ such that $d_F(v)=1$ for all $v\in X$ and $d_F(v)\ne 1$ for all $v\in Y$. Cornuéjols \cite{CHP} answered this question in the affirmative. Yu and Liu \cite{YL09} asked whether, for a given integer $k\ge 3$, every $k$-regular bipartite graph contains a 1-anti-factor. This paper answers this question in the affirmative.