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 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.

On which groups can arise as the canonical group of a spherical latin bitrade

“…The group A P has the 'universal' property that any minimal abelian representation of P is a quotient of A P , [11], also see [1]. Moreover A P is of the form Z⊕Z⊕C P , again see [1]. Drápal et al [10] and Cavenagh and Wanless [8] proved that C W is finite when (W, B) is a spherical latin bitrade.…”

“…The group A P has the 'universal' property that any minimal abelian representation of P is a quotient of A P , [11], also see [1]. Moreover A P is of the form Z⊕Z⊕C P , again see [1].…”

“…By defining φ| R = φ 1 , φ| C = φ 2 , and φ| S = −φ 3 it follows, see [1], that P embeds in an abelian group Γ if and only if there exists a function φ : R ⊔ C ⊔ S → Γ that is injective when restricted to each of R, C and S and is such that φ(r)+φ(c)+φ(s) = 0 for all (r, c, s) ∈ P . The map φ is called an embedding of P .…”

“…Given a face 2-coloured triangulation of the sphere in which the underlying graph is not necessarily simple and leaving the definitions of A W and A B unchanged it is still the case that [1]. In [18] the second author showed that given any finite abelian group Γ there exists a face 2-coloured triangulation of the sphere whose canonical group is isomorphic to Γ.…”

“…Note that given any of D R , D C or D S the original face two-coloured triangulation can be obtained by reversing the above construction: Tutte's Trinity Theorem [20] states that |S(D R )| = |S(D C )| = |S(D S )|. For a spherical latin bitrade (W, B) with corresponding face two-coloured triangulation G, this result was strengthened implicitly in [1] and explicitly in [18]…”