Antifactors of regular bipartite graphs

TL;DR

This paper proves that every k-regular bipartite graph contains a 1-anti-factor, answering a previously open question and extending understanding of factorization properties in bipartite graphs.

Contribution

The paper establishes that all k-regular bipartite graphs admit a 1-anti-factor, confirming a conjecture posed by Yu and Liu.

Findings

Every k-regular bipartite graph has a 1-anti-factor.

The result extends the class of graphs known to contain specific anti-factors.

Provides a positive answer to an open problem in graph factorization theory.

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 $G$. Lov\'asz 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 $d_F(v)\neq 1$ for all $v\in Y$. Cornu\'ejols \cite{CHP} answered this question in the affirmative. Yu and Liu \cite{YL09} asked whether, for a given integer $k\geq 3$, every $k$-regular bipartite graph contains a 1-anti-factor. This paper answers this question in the affirmative.