## 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.

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Blackburn, S.R., McCourt, T.A. Triangulations of the sphere, bitrades and abelian groups.
*Combinatorica* **34, **527–546 (2014). https://doi.org/10.1007/s00493-014-2924-7

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### Mathematics Subject Classification (2000)

- 05C10
- 05B07
- 05E99