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

TL;DR

This paper investigates conditions under which graphs contain even $[a,b]$-factors, providing counterexamples to a conjecture and establishing sharp sufficient conditions involving connectivity and eigenvalues.

Contribution

The paper offers counterexamples to Matsuda's conjecture and presents new sharp conditions for the existence of even $[a,b]$-factors in graphs.

Findings

Counterexamples are highly connected graphs that do not have the conjectured factors.

Sharp sufficient conditions are established for the existence of even $[a,b]$-factors.

A conjecture relating the largest eigenvalue to the existence of $[a,b]$-factors is proposed.

Abstract

Let $a$ and $b$ be positive 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 $a \le d_H(v) \le b$. Matsuda conjectured that if $G$ is an $n$-vertex 2-edge-connected graph such that $n \ge 2a+b+\frac{a^2-3a}b - 2$, $\delta(G) \ge a$, and $\sigma_2(G) \ge \frac{2an}{a+b}$, then $G$ has an even $[a,b]$-factor. In this paper, we provide counterexamples, which are highly connected. Furthermore, we give sharp sufficient conditions for a graph to have an even $[a,b]$-factor. For even $an$, we conjecture a lower bound for $\lambda_1(G)$ in an $n$-vertex graph to have an $[a,b]$-factor, where $\lambda_1(G)$ is the largest eigenvalue of $G$.