Vertex-deleted subgraphs and regular factors from regular graph

TL;DR

This paper investigates conditions under which the removal of any vertex from a specific class of regular, edge-connected graphs guarantees the existence of a k-factor in the resulting subgraph.

Contribution

It provides new sufficient conditions ensuring that vertex-deleted subgraphs of certain regular graphs contain a k-factor, extending previous results in graph factor theory.

Findings

Identifies conditions guaranteeing k-factors after vertex removal

Extends known results for regular, edge-connected graphs

Applicable to graphs with odd order and specific regularity

Abstract

Let $k$, $m$ and $r$ be three integers such that $2\leq k\leq m\leq r$. Let $G$ be a $2r$-regular, $2m$-edge-connected graph of odd order. We obtain some sufficient conditions, which guarantee $G-v$ contains a $k$-factor for all $v\in V(G)$.