Triangulations of the sphere, bitrades and abelian groups

Abstract

Let $\mathcal{G}$ be a triangulation of the sphere with vertex set V, such that the faces of the triangulation are properly coloured black and white. Motivated by applications in the theory of bitrades, Cavenagh and Wanless defined $\mathcal{A}_W$ to be the abelian group generated by the set V, with relations r+c+s = 0 for all white triangles with vertices r, c and s. The group $\mathcal{A}_B$ can be de fined similarly, using black triangles.

The paper shows that $\mathcal{A}_W$ and $\mathcal{A}_B$ are isomorphic, thus establishing the truth of a well-known conjecture of Cavenagh and Wanless. Connections are made between the structure of $\mathcal{A}_W$ and the theory of asymmetric Laplacians of finite directed graphs, and weaker results for orientable surfaces of higher genus are given. The relevance of the group $\mathcal{A}_W$ to the understanding of the embeddings of a partial latin square in an abelian group is also explained.