An odd $[1,b]$-factor in regular graphs from eigenvalues

Abstract

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 $r$.

6 pages