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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 $v \in V(G)$, $d_H(v)$ iseven and $a \le d_H(v) \le b$. Matsuda conjectured that if $G$ is an $n$-vertex2-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 highlyconnected. Furthermore, we give sharp sufficient conditions for a graph to havean 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 $G$.

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