{
  "id": "1601.05040",
  "version": "v1",
  "published": "2016-01-19T19:24:18.000Z",
  "updated": "2016-01-19T19:24:18.000Z",
  "title": "Maximizing $H$-colorings of connected graphs with fixed minimum degree",
  "authors": [
    "John Engbers"
  ],
  "comment": "9 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "For graphs $G$ and $H$, an $H$-coloring of $G$ is a map from the vertices of $G$ to the vertices of $H$ that preserves edge adjacency. We consider the following extremal enumerative question: for a given $H$, which connected $n$-vertex graph with minimum degree $\\delta$ maximizes the number of $H$-colorings? We show that for non-regular $H$ and sufficiently large $n$, the complete bipartite graph $K_{\\delta,n-\\delta}$ is the unique maximizer. As a corollary, for non-regular $H$ and sufficiently large $n$ the graph $K_{k,n-k}$ is the unique $k$-connected graph that maximizes the number of $H$-colorings among all $k$-connected graphs. Finally, we show that this conclusion does not hold for all regular $H$ by exhibiting a connected $n$-vertex graph with minimum degree $\\delta$ which has more $K_{q}$-colorings (for sufficiently large $q$ and $n$) than $K_{\\delta,n-\\delta}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-19T19:24:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "05C35",
      "05C40"
    ],
    "keywords": [
      "connected graph",
      "fixed minimum degree",
      "sufficiently large",
      "vertex graph",
      "complete bipartite graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160105040E"
    }
  }
}