{
  "id": "1809.01259",
  "version": "v1",
  "published": "2018-09-04T22:32:24.000Z",
  "updated": "2018-09-04T22:32:24.000Z",
  "title": "Sidorenko's conjecture for blow-ups",
  "authors": [
    "David Conlon",
    "Joonkyung Lee"
  ],
  "comment": "12 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "A celebrated conjecture of Sidorenko and Erd\\H{o}s-Simonovits states that, for all bipartite graphs $H$, quasirandom graphs contain asymptotically the minimum number of copies of $H$ taken over all graphs with the same order and edge density. This conjecture has attracted considerable interest over the last decade and is now known to hold for a broad range of bipartite graphs, with the overall trend saying that a graph satisfies the conjecture if it can be built from simple building blocks such as trees in a certain recursive fashion. Our contribution here, which goes beyond this paradigm, is to show that the conjecture holds for any bipartite graph $H$ with bipartition $A \\cup B$ where there are many vertices in $B$ whose degree equals the maximum degree. As a corollary, we have that for every bipartite graph $H$ with bipartition $A \\cup B$, there is a positive integer $p$ such that the blow-up $H_A^p$ formed by taking $p$ vertex-disjoint copies of $H$ and gluing all copies of $A$ along corresponding vertices satisfies the conjecture. Another way of viewing this latter result is that for every bipartite $H$ there is a positive integer $p$ such that an $L^p$-version of Sidorenko's conjecture holds for $H$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-04T22:32:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bipartite graph",
      "quasirandom graphs contain",
      "positive integer",
      "sidorenkos conjecture holds",
      "bipartition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}