Dataset of Edmonds' bi-vectors and tri-vectors with realizations.

In 1965, Jack Edmonds characterized pairs of graphs and 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, …, d) and = (t,…, t) denote the degree sequences of two such graphs, then , where is the number of edges in each of the two graphs and is the Euler characteristic of the surface. However, this condition is not sufficient, and it is an open question to characterize bi-vectors () that are , that is, that can be realized as the degree sequences of pairs and of surface-embedded graphs. The above question is a special case of the following one. A multigraph is even if each vertex has even degree and 3-colored if is equipped with a fixed proper coloring of its vertex set assigning each vertex a color in the set {1,2,3}. Let be a 3-colored even multigraph embedded in a surface so that every face is a triangle. Denote by = (d, …, d), = (t, …, t), and = (δ, ..., …, δ) the sequences of half-degrees of vertices of of colors 1, 2, and 3, respectively. Then, , where is the Euler characteristic of the surface A tri-vector satisfying the above conditions is called . A feasible tri-vector is called 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.