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DOI: 10.4230/LIPIcs.APPROX-RANDOM.2019.36

On a Connectivity Threshold for Colorings of Random Graphs and Hypergraphs

Authors: Michael Anastos and Alan Frieze

Published in: LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

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.

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Michael Anastos and Alan Frieze. On a Connectivity Threshold for Colorings of Random Graphs and Hypergraphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 36:1-36:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{anastos_et_al:LIPIcs.APPROX-RANDOM.2019.36,
  author =	{Anastos, Michael and Frieze, Alan},
  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 =	{Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.36},
  URN =		{urn:nbn:de:0030-drops-112513},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.36},
  annote =	{Keywords: Random Graphs, Colorings, Ergodicity}
}

Document

RANDOM

DOI: 10.4230/LIPIcs.APPROX-RANDOM.2019.66

Thresholds in Random Motif Graphs

Authors: Michael Anastos, Peleg Michaeli, and Samantha Petti

Published in: LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

Abstract

We introduce a natural generalization of the Erdős-Rényi random graph model in which random instances of a fixed motif are added independently. The binomial random motif graph G(H,n,p) is the random (multi)graph obtained by adding an instance of a fixed graph H on each of the copies of H in the complete graph on n vertices, independently with probability p. We establish that every monotone property has a threshold in this model, and determine the thresholds for connectivity, Hamiltonicity, the existence of a perfect matching, and subgraph appearance. Moreover, in the first three cases we give the analogous hitting time results; with high probability, the first graph in the random motif graph process that has minimum degree one (or two) is connected and contains a perfect matching (or Hamiltonian respectively).

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Michael Anastos, Peleg Michaeli, and Samantha Petti. Thresholds in Random Motif Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 66:1-66:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{anastos_et_al:LIPIcs.APPROX-RANDOM.2019.66,
  author =	{Anastos, Michael and Michaeli, Peleg and Petti, Samantha},
  title =	{{Thresholds in Random Motif Graphs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{66:1--66:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-125-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{145},
  editor =	{Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.66},
  URN =		{urn:nbn:de:0030-drops-112819},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.66},
  annote =	{Keywords: Random graph, Connectivity, Hamiltonicty, Small subgraphs}
}

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On a Connectivity Threshold for Colorings of Random Graphs and Hypergraphs

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Thresholds in Random Motif Graphs

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