Random Embeddings of Graphs: The Expected Number of Faces in Most Graphs is Logarithmic

A random 2-cell embedding of a connected graph G in some orientable surface is obtained by choosing a random local rotation around each vertex. Under this setup, the number of faces or the genus of the corresponding 2-cell embedding becomes a random variable. Random embeddings of two particular graph classes - those of a bouquet of n loops and those of n parallel edges connecting two vertices - have been extensively studied and are well-understood. However, little is known about more general graphs despite their important connections with central problems in mainstream mathematics and in theoretical physics (see [Lando & Zvonkin, Graphs on surfaces and their applications, Springer 2004]). There are also tight connections with problems in computing (random generation, approximation algorithms). The results of this paper, in particular, explain why Monte Carlo methods (see, e.g., [Gross & Tucker, Local maxima in graded graphs of imbeddings, Ann. NY Acad. Sci 1979] and [Gross & Rieper, Local extrema in genus stratified graphs, JGT 1991]) cannot work for approximating the minimum genus of graphs.

In his breakthrough work ([Stahl, Permutation-partition pairs, JCTB 1991] and a series of other papers), Stahl developed the foundation of “random topological graph theory”. Most of his results have been unsurpassed until today. In our work, we analyze the expected number of faces of random embeddings (equivalently, the average genus) of a graph G. It was very recently shown [Campion Loth & Mohar, Expected number of faces in a random embedding of any graph is at most linear, CPC 2023] that for any graph G, the expected number of faces is at most linear. We show that the actual expected number of faces F(G) is almost always much smaller. In particular, we prove the following results:

(1) ½ ln n - 2 < 𝔼 [F(K_n)] ≤ 3.65 ln n+ o(1). This substantially improves Stahl's n + ln n upper bound for this case.

(2) For random graphs G(n,p) (p = p(n)), we have .

(3) For random models B(n, Δ) containing only graphs, whose maximum degree is at most Δ, we obtain stronger bounds by showing that the expected number of faces is Θ(ln n).

^* The full version of our paper available on arXiv https://arxiv.org/abs/2211.01032.