{
  "id": "1506.05388",
  "version": "v1",
  "published": "2015-06-17T17:27:22.000Z",
  "updated": "2015-06-17T17:27:22.000Z",
  "title": "Extremal H-colorings of trees and 2-connected graphs",
  "authors": [
    "John Engbers",
    "David Galvin"
  ],
  "comment": "14 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "For graphs $G$ and $H$, an $H$-coloring of $G$ is an adjacency preserving map from the vertices of $G$ to the vertices of $H$. $H$-colorings generalize such notions as independent sets and proper colorings in graphs. There has been much recent research on the extremal question of finding the graph(s) among a fixed family that maximize or minimize the number of $H$-colorings. In this paper, we prove several results in this area. First, we find a class of graphs ${\\mathcal H}$ with the property that for each $H \\in {\\mathcal H}$, the $n$-vertex tree that minimizes the number of $H$-colorings is the path $P_n$. We then present a new proof of a theorem of Sidorenko, valid for large $n$, that for every $H$ the star $K_{1,n-1}$ is the $n$-vertex tree that maximizes the number of $H$-colorings. Our proof uses a stability technique which we also use to show that for any non-regular $H$ (and certain regular $H$) the complete bipartite graph $K_{2,n-2}$ maximizes the number of $H$-colorings of $n$-vertex $2$-connected graphs. Finally, we show that the cycle $C_n$ maximizes the number of proper colorings of $n$-vertex $2$-connected graphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-17T17:27:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C05",
      "05C15",
      "05C35"
    ],
    "keywords": [
      "extremal h-colorings",
      "vertex tree",
      "proper colorings",
      "connected graphs",
      "complete bipartite graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150605388E"
    }
  }
}