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DOI: 10.4230/LIPIcs.APPROX-RANDOM.2019.36
URN: urn:nbn:de:0030-drops-112513
URL: https://drops.dagstuhl.de/opus/volltexte/2019/11251/
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Anastos, Michael ; Frieze, Alan

On a Connectivity Threshold for Colorings of Random Graphs and Hypergraphs

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Abstract

Let Omega_q=Omega_q(H) denote the set of proper [q]-colorings of the hypergraph H. Let Gamma_q be the graph with vertex set Omega_q where two vertices are adjacent iff the corresponding colorings differ in exactly one vertex. We show that if H=H_{n,m;k}, k >= 2, the random k-uniform hypergraph with V=[n] and m=dn/k hyperedges then w.h.p. Gamma_q is connected if d is sufficiently large and q >~ (d/log d)^{1/(k-1)}. This is optimal to the first order in d. Furthermore, with a few more colors, we find that the diameter of Gamma_q is O(n) w.h.p, where the hidden constant depends on d. So, with this choice of d,q, the natural Glauber Dynamics Markov Chain on Omega_q is ergodic w.h.p.

BibTeX - Entry

@InProceedings{anastos_et_al:LIPIcs:2019:11251,
  author =	{Michael Anastos and Alan Frieze},
  title =	{{On a Connectivity Threshold for Colorings of Random Graphs and Hypergraphs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{36:1--36:10},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-125-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{145},
  editor =	{Dimitris Achlioptas and L{\'a}szl{\'o} A. V{\'e}gh},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/11251},
  URN =		{urn:nbn:de:0030-drops-112513},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.36},
  annote =	{Keywords: Random Graphs, Colorings, Ergodicity}
}

Keywords: Random Graphs, Colorings, Ergodicity
Seminar: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)
Issue Date: 2019
Date of publication: 19.09.2019


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