  An odd $[1,b]$-factor of a graph $G$ is a spanning subgraph $H$ such that for
each vertex $v \in V(G)$, $d_H(v)$ is odd and $1\le d_H(v) \le b$. Let
$\lambda_3(G)$ be the third largest eigenvalue of the adjacency matrix of $G$.
For positive integers $r \ge 3$ and even $n$, Lu, Wu, and Yang [10] proved a
lower bound for $\lambda_3(G)$ in an $n$-vertex $r$-regular graph $G$ to
gurantee the existence of an odd $[1,b]$-factor in $G$. In this paper, we
improve the bound; it is sharp for every $r$.

Combinatorics