The evolution of random graphs on surfaces

Abstract

Abstract. For integers g, m ≥ 0 and n > 0, let Sg(n, m) denote the graph taken uniformly at random from the set of all graphs on {1, 2, . . . , n} with exactly m = m(n) edges and with genus at most g. We use counting arguments to investigate the components, subgraphs, maximum degree, and largest face size of Sg(n, m), finding that there is often different asymptotic behaviour depending on the ratio m n . In our main results, we show that the probability that Sg(n, m) contains any given non-planar component converges to 0 as n → ∞ for all m(n); the probability that Sg(n, m) contains a copy of any given planar graph converges to 1 as n → ∞ if lim inf m n > 1; the maximum degree of Sg(n, m) is Θ(ln n) with high probability if lim inf m n > 1; and the largest face size of Sg(n, m) has a threshold around m n = 1 where it changes from Θ(n) to Θ(ln n) with high probability.

Concentration of maximum degree in random planar graphs

“…Using Lemma 2.11(g) we obtain ν (n, 2m) = ν (n) + o (1). Together with (11) this shows the statement.…”

Longest and shortest cycles in random planar graphs

“…Using (7) and the fact that s = o(n) we get n U = (1 + o(1))n. Combining that with (10) yields that for n large enough…”

Random cographs: Brownian graphon limit and asymptotic degree distribution

“…Random graphs are arguably the most studied objects at the interface of combinatorics and probability theory. One aspect of their study consists in analyzing a uniform random graph of large size n in a prescribed family, for example, perfect graphs [31], planar graphs [32], graphs embeddable in a surface of given genus [19], graphs in subcritical classes [35], hereditary classes [23], or addable classes [13,30]. The present paper focuses on uniform random cographs (both in the labeled and unlabeled settings).…”