A Note On Edge Connectivity and Parity Factor

TL;DR

This paper explores conditions for the existence of parity factors in graphs, focusing on minimum degree restrictions and edge connectivity, using Lovász's (g,f)-parity theorem to establish necessary and sufficient conditions.

Contribution

It provides new necessary and sufficient conditions for graphs to have parity factors with specific minimum degrees based on edge connectivity, advancing graph factor theory.

Findings

Characterization of graphs with parity factors with restricted minimum degree

Sufficient conditions for parity factors with minimum degree m based on edge connectivity

Application of Lovász's (g,f)-parity theorem to derive these conditions

Abstract

In this paper, we investigate some parity factors by using Lov\'asz's (g,f)-parity theorem. Let $m>0$ be an integer. Firstly, we obtain a sufficient and necessary condition for some graphs to have a parity factor with restricted minimum degree. Secondly, we obtain some sufficient conditions for a graph to have a parity factor with minimum degree $m$ in term of edge connectivity.