A Degree Condition for a Graph to have $(a,b)$-Parity Factors

TL;DR

This paper establishes a new degree condition under which a graph contains an $(a,b)$-parity factor, extending previous results and demonstrating the tightness of these conditions.

Contribution

It introduces a novel degree criterion ensuring the existence of $(a,b)$-parity factors, generalizing and extending prior work on $k$-factors.

Findings

Provides a sufficient degree condition for $(a,b)$-parity factors.

Extends Nishimura's results on $k$-factors.

Generalizes Li and Cai's findings with tight conditions.

Abstract

Let $a,b,n$ be three positive integers such that $a\equiv b\pmod 2$ and $n\geq b(a+b)(a+b+2)/(2a)$. Let $G$ be a graph of order $n$ with minimum degree at least $a+b/a-1$. We show that $G$ has an $(a,b)$-parity factor, if $max\{d_G(u),d_G(v)\}\geq \frac{an}{a+b}$ for any two nonadjacent vertices $u,v$ of $G$. It is an extension of Nishimura's results for the existence of $k$-factors (\emph{J. Graph Theory}, \textbf{16} (1992), 141--151) and generalizes Li and Cai's result in some senses (\emph{J. Graph Theory}, \textbf{27} (1998), 1--6). These conditions are tight.