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Thresholds in Random Motif Graphs

Authors Michael Anastos , Peleg Michaeli , Samantha Petti



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Author Details

Michael Anastos
  • Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania, USA
Peleg Michaeli
  • School of Mathematical Sciences, Tel Aviv University, Israel
Samantha Petti
  • School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia, USA

Acknowledgements

We thank Alan Frieze for helpful discussions and for connecting the authors.

Cite AsGet BibTex

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)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.66

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).

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Random graphs
  • Mathematics of computing → Paths and connectivity problems
Keywords
  • Random graph
  • Connectivity
  • Hamiltonicty
  • Small subgraphs

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