Spectral versions on Lovász’s (a,b)-parity factor theorem in graphs

Abstract

Let $G$ be a graph, and let $g$ and $f$ be two integer-valued functions defined on $V\left(G\right)$ such that $0\le g\left(v\right)\le f\left(v\right)$ for every $v\in V\left(G\right).$ A $(g,f)$-parity factor of $G$ is a spanning subgraph $H$ of $G$ such that $d_H\left(v\right)\equiv g\left(v\right)\equiv f\left(v\right)\left(\mathrm{mod}2\right)$ and $g\left(v\right)\le d_H\left(v\right)\le f\left(v\right)$ for every $v\in V\left(G\right).$ In particular, a $(g,f)$-parity factor is called an $(a,b)$-parity factor if $g\left(v\right)\equiv a$ and $f\left(v\right)\equiv b,$ where $a$ and $b$ are two positive integers satisfying $a\le b$ and $d_H\left(v\right)\equiv a\equiv b\left(\mathrm{mod}2\right).$ In recent years, many interesting researches focus on establishing sufficient conditions to ensure that a graph contains an $(a,b)$-parity factor. Based on Lovász’s $(a,b)$-parity factor theorem and technical distance spectral methods, which are very different from the adjacency spectrum, we in this paper present a sufficient condition in terms of the distance spectral radius for a graph to contain an $(a,b)$-parity factor. Moreover, we also prove the $Q$-spectral version of Lovász’s $(a,b)$-parity factor theorem in graphs.