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

An odd [ 1 , b ] -factor of a graph G is a spanning subgraph H such that for each vertex v ∈ V ( G ) , d H ( v ) is odd and 1 ≤ d H ( v ) ≤ b . Let λ 3 ( G ) be the third largest eigenvalue of the adjacency matrix of G . For positive integers r ≥ 3 and even n , Lu et al. (2010) proved a lower bound for λ 3 ( G ) in an n -vertex r -regular graph G to guarantee the existence of an odd [ 1 , b ] -factor in G . In this paper, we improve the bound; it is sharp for every r .