LIPIcs - Leibniz International Proceedings in Informatics. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)
We introduce a natural generalization of the Erd\H{o}s-R\'enyi 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).Comment: 19 page