Triangulations of the sphere, bitrades and abelian groups

TL;DR

This paper proves that two abelian groups derived from sphere triangulations with face colorings are isomorphic, confirming a conjecture, and explores their connections to graph Laplacians and Latin square embeddings.

Contribution

It establishes the isomorphism between the groups $A_W$ and $A_B$, confirming a conjecture, and links these groups to graph Laplacians and Latin square embeddings.

Findings

$A_W$ and $A_B$ are isomorphic for sphere triangulations.

Connections between $A_W$ and asymmetric Laplacians are established.

Weaker results are obtained for higher genus surfaces.

Abstract

Let $G$ 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 $A_W$ 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 $A_B$ can be defined similarly, using black triangles. The paper shows that $A_W$ and $A_B$ are isomorphic, thus establishing the truth of a well-known conjecture of Cavenagh and Wanless. Connections are made between the structure of $A_W$ 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 $A_W$ to the understanding of the embeddings of a partial latin square in an abelian group is also explained.