# Journal of theKorean Mathematical SocietyJKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc.

Published online April 5, 2022

## Sharp Ore-type conditions for the existence of an even $[4,b]$-factor in a graph

Eun-Kyung Cho, Su-Ah Kwon, and Suil O

Hankuk University of Foreign Studies, SUNY-Korea

### Abstract

Let $a$ and $b$ be positive even integers. An even $[a,b]$-factor of a graph $G$ is a spanning subgraph $H$ such that for every vertex $v \in V(G)$, $d_H(v)$ is even and $a \le d_H(v) \le b$. Let $\kappa(G)$ be the minimum size of a vertex set $S$ such that $G-S$ is disconnected or one vertex, and let $\sigma_2(G)=\min_{uv \notin E(G)}(d(u)+d(v))$. In 2005, Matsuda proved an Ore-type condition for an $n$-vertex graph satisfying certain properties to guarantee the existence of an even $[2,b]$-factor.
In this paper, we prove that for an even positive integer $b$ with $b \ge 6$, if $G$ is an $n$-vertex graph such that $n \ge b+5$, $\kappa(G) \ge 4$, and $\sigma_2(G) \ge \frac{8n}{b+4}$, then $G$ contains an even $[4,b]$-factor;
each condition on $n$, $\kappa(G)$, and $\sigma_2(G)$ is sharp.

Keywords: Even $[4,b]$-factor, Ore-type condition, connectivity

MSC numbers: 05C70