The Evolution of Random Graphs on Surfaces

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.

Random graphs embeddable in order‐dependent surfaces

“…Thus (18) holds for n sufficiently large, and this completes the proof. ▪ 7 PROOF OF THEOREM 2, PROBABILITY THAT R n ∈ u 𝒜 g IS CONNECTED…”

“…Proof of Theorem 2(c). Suppose that g(n) ≫ n. By (20) and Lemma 39, we want to show that E[e ( Rn−1 ) ] ≫ n; and to show this, it suffices to show that E[e(R n )] ≫ n (since ĝ(n) = g(n + 1) ≫ n). Let g 1 (n) be any genus function such that n ≪ g 1 (n) ≪ n 2 , say g 1 (n) = ⌊n 3∕2 ⌋.…”

The genus of the Erdős‐Rényi random graph and the fragile genus property

“…One of our motivations for this paper comes from recent work concerning random graphs on given surfaces. The typical properties of graphs with genus at most g have been studied in and for the case when g is a fixed constant, and questions have been posed on the likely behavior when g is allowed to grow with n . Hence, in this section, we shall discuss the contiguity (see Definition ) of such random graph models with G ( n ) and G ( n , m ).…”

“…The genus is one of the most fundamental properties of a graph, and plays an important role in a number of applications and algorithms (e.g., coloring problems [29] and the manufacture of electrical circuits [12,24]). It is naturally intriguing to consider the genus of a random graph, and such matters are also related to random graphs on surfaces (see, e.g., Question 8.13 of [18] and Section 9 of [8]). In addition, results on the genus of random bipartite graphs [16] were recently used to provide a polynomial-time approximation scheme for the genus of dense graphs [15].…”

Classes of graphs embeddable in order-dependent surfaces

“…A central aim in both of these papers is to find where there is a change between 'planar-like' behaviour and behaviour like that of a binomial (Erdős-Rényi) random graph, both for class size and for typical properties. It seems that this 'phase transition' occurs when g(n) is around n/ log n. See [DKMS19] for results on the evolution of random graphs on non-constant orientable surfaces when we consider also the number of edges.…”