On the degree sequences of dual graphs on surfaces

TL;DR

This paper investigates the conditions under which degree sequences can be realized by dual graphs embedded on surfaces, extending Edmonds' criterion and identifying exceptions for certain surfaces.

Contribution

It provides new infinite series of exceptions for sphere and projective plane embeddings and conjectures no exceptions for surfaces with non-positive Euler characteristic.

Findings

Identified infinite exceptions for sphere and projective plane cases.

Used Edmonds' criterion to analyze realizability of degree sequences.

Conjectured no exceptions exist for surfaces with Euler characteristic ≤ 0.

Abstract

Given two graphs $G$ and $G^*$ with a one-to-one correspondence between their edges, when do $G$ and $G^*$ form a pair of dual graphs realizing the vertices and countries of a map embedded in a surface? A criterion was obtained by Jack Edmonds in 1965. Furthermore, let $\boldsymbol{d}=(d_1,\ldots,d_n)$ and $\boldsymbol{t}=(t_1,\ldots,t_m)$ be their degree sequences. Then, clearly, $\sum_{i=1}^n d_i = \sum_{j=1}^m t_j = 2\ell$, where $\ell$ is the number of edges in each of the two graphs, and $\chi = n - \ell + m$ is the Euler characteristic of the surface. Which sequences $\boldsymbol{d}$ and $\boldsymbol{t}$ satisfying these conditions still cannot be realized as the degree sequences? We make use of Edmonds' criterion to obtain several infinite series of exceptions for the sphere, $\chi = 2$, and projective plane, $\chi = 1$. We conjecture that there exist no exceptions for $\chi \leq 0$.