A spectral condition for the existence of cycles with consecutive odd lengths in non-bipartite graphs

Abstract

A graph G is called H-free, if it does not contain H as a subgraph. In 2010, Nikiforov proposed a Brualdi-Solheid-Turán type problem: what is the maximum spectral radius of an H-free graph of order n? In this paper, we consider the Brualdi-Solheid-Turán type problem for non-bipartite graphs. Let Ka,b•K3 denote the graph obtained by identifying a vertex of Ka,b in the part of size b and a vertex of K3. We prove that if G is a non-bipartite graph of order n satisfying ρ(G)≥ρ(K⌈n−22⌉,⌊n−22⌋•K3), then G contains all odd cycles C2l+1 for each integer l∈[2,k] unless G≅K⌈n−22⌉,⌊n−22⌋•K3, provided that n is sufficiently large with respect to k. This resolves the problem posed by Guo, Lin and Zhao (2021).