Eigenvalues and [a,b]‐factors in regular graphs

Abstract

For positive integers, r ≥ 3, h ≥ 1, and k ≥ 1, Bollobás, Saito, and Wormald proved some sufficient conditions for an h‐edge‐connected r‐regular graph to have a k‐factor in 1985. Lu gave an upper bound for the third largest eigenvalue in a connected r‐regular graph to have a k‐factor in 2010. Gu found an upper bound for certain eigenvalues in an h‐edge‐connected r‐regular graph to have a k‐factor in 2014. For positive integers a ≤ b, an even (or odd) [ a , b ]‐factor of a graph G is a spanning subgraph H such that for each vertex v ∈ V ( G ), d H ( v ) is even (or odd) and a ≤ d H ( v ) ≤ b. In this paper, we prove upper bounds (in terms of a , b , and r) for certain eigenvalues (in terms of a , b , r , and h) in an h‐edge‐connected r‐regular graph G to guarantee the existence of an even [ a , b ]‐factor or an odd [ a , b ]‐factor. This result extends the one of Bollbás, Saito, and Wormald, the one of Lu, and the one of Gu.