Asymptotics of local face distributions and the face distribution of the complete graph

TL;DR

This paper investigates the distribution of faces in graph embeddings, establishing asymptotic uniformity at vertices with large degrees, and applies this to understand face distributions in complete graphs, revealing connections to permutation sets.

Contribution

It introduces asymptotic uniformity results for local face distributions in large-degree graphs and links these to the global face distribution of complete graphs, including new character bounds.

Findings

Asymptotic uniformity of local face distribution for large-degree graphs

Local face distribution determines global face distribution in complete graphs

A large subset of the complete graph shares face distribution properties with permutations

Abstract

We are interested in the distribution of the number of faces across all the $2-$cell embeddings of a graph, which is equivalent to the distribution of genus by Euler's formula. In order to study this distribution, we consider the local distribution of faces at a single vertex. We show an asymptotic uniformity on this local face distribution which holds for any graph with large vertex degrees. We use this to study the usual face distribution of the complete graph. We show that in this case, the local face distribution determines the face distribution for almost all of the whole graph. We use this result to show that a portion of the complete graph of size $(1-o(1))|K_n|$ has the same face distribution as the set of all permutations, up to parity. Along the way, we prove new character bounds and an asymptotic uniformity on conjugacy class products.