Thresholds in Random Motif Graphs

Authors Michael Anastos , Peleg Michaeli , Samantha Petti

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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


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

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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)


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
  • Random graph
  • Connectivity
  • Hamiltonicty
  • Small subgraphs


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