# Dataset of Edmonds’ bi-vectors and tri-vectors with realizations

## TL;DR

This paper introduces a dataset of geographic bi-vectors and tri-vectors, which represent degree sequences of dual graphs embedded on surfaces.

## Contribution

The novelty is the creation of a dataset of geographic bi-vectors and tri-vectors with explicit realizations.

## Key findings

- A dataset of geographic bi-vectors and tri-vectors is presented.
- Each entry includes realizations proving their geographic nature.

## Abstract

In 1965, Jack Edmonds characterized pairs of graphs G and G* with a bijection between their edge sets that form a pair of dual graphs realizing the vertices and countries of a map embedded in a surface. A necessary condition is that, if d = (d1, …, dn) and t = (t1,…, tm) denote the degree sequences of two such graphs, then ∑i=1ndi=∑j=1mtj=2l, where l is the number of edges in each of the two graphs and χ=n+m−l is the Euler characteristic of the surface. However, this condition is not sufficient, and it is an open question to characterize bi-vectors (d, t) that are geographic, that is, that can be realized as the degree sequences of pairs G and G* of surface-embedded graphs.

The above question is a special case of the following one. A multigraph G is even if each vertex has even degree and 3-colored if G is equipped with a fixed proper coloring of its vertex set assigning each vertex a color in the set {1,2,3}. Let G be a 3-colored even multigraph embedded in a surface S so that every face is a triangle. Denote by d = (d1, …, dn), t = (t1, …, tm), and δ = (δ1, ..., …, δk) the sequences of half-degrees of vertices of G of colors 1, 2, and 3, respectively. Then, ∑i=1ndi=∑j=1mtj=∑μ=1ktμ=l, where χ=n+k+m−l is the Euler characteristic of the surface S. A tri-vector (d, t, δ) satisfying the above conditions is called feasible. A feasible tri-vector is called geographic if it is realized by a 3-colored triangulation of a surface. Geographic tri-vectors extend the concept of geographic bi-vectors.

We present a dataset of geographic bi-vectors and tri-vectors, along with realizations proving that they are geographic.