On the local genus distribution of graph embeddings

TL;DR

This paper investigates how reembedding a vertex in a graph's surface embedding affects the genus, providing formulas and conditions to understand minimal genus configurations and probabilities of one-face embeddings.

Contribution

It introduces formulas for counting reembeddings that change genus and offers criteria for minimal genus embeddings, advancing understanding of graph embeddings on surfaces.

Findings

Formulas to compute reembedding counts for genus changes

A lower bound on the probability of one-face embeddings

A necessary condition for minimal genus embeddings

Abstract

The $2$-cell embeddings of graphs on closed surfaces have been widely studied. It is well known that ($2$-cell) embedding a given graph $G$ on a closed orientable surface is equivalent to cyclically ordering the edges incident to each vertex of $G$. In this paper, we study the following problem: given a genus $g$ embedding $\epsilon$ of the graph $G$ and a vertex of $G$, how many different ways of reembedding the vertex such that the resulting embedding $\epsilon'$ is of genus $g+\Delta g$? We give formulas to compute this quantity and the local minimal genus achieved by reembedding. In the process we obtain miscellaneous results. In particular, if there exists a one-face embedding of $G$, then the probability of a random embedding of $G$ to be one-face is at least $\prod_{\nu\in V(G)}\frac{2}{deg(\nu)+2}$, where $deg(\nu)$ denotes the vertex degree of $\nu$. Furthermore we obtain an easy-to-check necessary condition for a given embedding of $G$ to be an embedding of minimum genus.