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# Thresholds in Random Motif Graphs

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LIPIcs.APPROX-RANDOM.2019.66.pdf
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## Acknowledgements

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

## Cite As

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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