Triangulations of the sphere, bitrades and abelian groups
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 wellknown 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2011
 arXiv:
 arXiv:1112.5423
 Bibcode:
 2011arXiv1112.5423B
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Group Theory;
 05B07
 EPrint:
 22 pages