{
  "id": "1304.1680",
  "version": "v2",
  "published": "2013-04-05T11:12:14.000Z",
  "updated": "2013-05-14T10:20:38.000Z",
  "title": "Degree powers in $C_5$-free graphs",
  "authors": [
    "Ran Gu",
    "Xueliang Li",
    "Yongtang Shi"
  ],
  "comment": "10 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G$ be a graph with degree sequence $d_1,d_2,\\ldots,d_n$. Given a positive integer $p$, denote by $e_p(G)=\\sum_{i=1}^n d_i^p$. Caro and Yuster introduced a Tur\\'an-type problem for $e_p(G)$: given an integer $p$, how large can $e_p(G)$ be if $G$ has no subgraph of a particular type. They got some results for the subgraph of particular type to be a clique of order $r+1$ and a cycle of even length, respectively. Denote by $ex_p(n,H)$ the maximum value of $e_p(G)$ taken over all graphs with $n$ vertices that do not contain $H$ as a subgraph. Clearly, $ex_1(n,H)=2ex(n,H)$, where $ex(n,H)$ denotes the classical Tur\\'an number. In this paper, we consider $ex_p(n, C_5)$ and prove that for any positive integer $p$ and sufficiently large $n$, there exists a constant $c=c(p)$ such that the following holds: if $ex_p(n, C_5)=e_p(G)$ for some $C_5$-free graph $G$ of order $n$, then $G$ is a complete bipartite graph having one vertex class of size $cn+o(n)$ and the other $(1-c)n+o(n)$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-05-14T10:20:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C35",
      "05C07"
    ],
    "keywords": [
      "free graph",
      "degree powers",
      "positive integer",
      "complete bipartite graph",
      "vertex class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.1680G"
    }
  }
}