{
  "id": "2110.11345",
  "version": "v2",
  "published": "2021-10-20T15:53:50.000Z",
  "updated": "2022-07-19T04:49:02.000Z",
  "title": "A spectral condition for the existence of cycles with consecutive odd lengths in non-bipartite graphs",
  "authors": [
    "Zhiyuan Zhang",
    "Yanhua Zhao"
  ],
  "comment": "13 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "A graph $G$ is called $H$-free, if it does not contain $H$ as a subgraph. In 2010, Nikiforov proposed a Brualdi-Solheid-Tur\\'{a}n type problem: what is the maximum spectral radius of an $H$-free graph of order $n$? In this paper, we consider the Brualdi-Solheid-Tur\\'{a}n type problem for non-bipartite graphs. Let $K_{a, b}\\bullet K_3$ denote the graph obtained by identifying a vertex of $K_{a,b}$ in the part of size $b$ and a vertex of $K_3$. We prove that if $G$ is a non-bipartite graph of order $n$ satisfying $\\rho(G)\\geq \\rho(K_{\\lceil\\frac{n-2}{2}\\rceil, \\lfloor\\frac{n-2}{2}\\rfloor}\\bullet K_3)$, then $G$ contains all odd cycles $C_{2l+1}$ for each integer $l\\in[2,k]$ unless $G\\cong K_{\\lceil\\frac{n-2}{2}\\rceil, \\lfloor\\frac{n-2}{2}\\rfloor}\\bullet K_3$, provided that $n$ is sufficiently large with respect to $k$. This resolves the problem posed by Guo, Lin and Zhao (2021).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-07-19T04:49:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50"
    ],
    "keywords": [
      "non-bipartite graph",
      "consecutive odd lengths",
      "spectral condition",
      "type problem",
      "maximum spectral radius"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}