Phase transitions in graphs on orientable surfaces

Abstract

Let S g be the orientable surface of genus g and denote by  g (n, m) the class of all graphs on vertex set [n] = {1, … , n} with m edges embeddable on S g . We prove that the component structure of a graph chosen uniformly at random from  g (n, m) features two phase transitions. The first phase transition mirrors the classical phase transition in the Erdős-Rényi random graph G(n, m) chosen uniformly at random from all graphs with vertex set [n] and m edges. It takes place at m = n 2 + O(n 2∕3 ), when the giant component emerges. The second phase transition occurs at m = n + O(n 3∕5 ), when the giant component covers almost all vertices of the graph. This kind of phenomenon is strikingly different from G(n, m) and has only been observed for graphs on surfaces.

The Evolution of Random Graphs on Surfaces

“…We start by stating two very useful ingredients (Lemma 2.10 and Lemma 2.11) concerning the number of appearances and triangulated appearances (the proofs of which will be given in Sections 7 and 8): The following result, recently given in [24] (see also [23]), will be of use to us as well (note that the brace notation used here is to be understood as meaning that both conditions must be satisfied simultaneously): We note that it is not known whether the o n (log n) 2/3 condition in Lemma 2.12 can be amended to o(n).…”

Longest and shortest cycles in random planar graphs

“…Furthermore, it is known that the "edge density" in the part without the largest component is typically much larger in P(n, m) than in G(n, m) (see, e.g., [23,Theorem 1.7]). This affects the order of the longest cycle outside the largest component (cf.…”

“…Thus, they are kernel-stable classes due to Lemma 2. Moreover, in [23], it was shown that the class of graphs that are embeddable on an orientable surface of genus g ∈ N ∪ {0} satisfies (P2). Thus, they also form a kernel-stable class of graphs, since they trivially fulfill (P1).…”

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 and the manufacture of electrical circuits ). 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 and Section 9 of ). In addition, results on the genus of random bipartite graphs were recently used to provide a polynomial‐time approximation scheme for the genus of dense graphs .…”