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

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 $G$.

13 pages, 2 figures