Latin trades in groups defined on planar triangulations

Cavenagh, Nicholas J. ; Wanless, Ian M.

Journal of Algebraic Combinatorics, Tome 30 (2009) no. 3, pp. 323-347.

Résumé

Summary: For a finite triangulation of the plane with faces properly coloured white and black, let $A _{ W}$ mathcalA_W be the abelian group constructed by labelling the vertices with commuting indeterminates and adding relations which say that the labels around each white triangle add to the identity. We show that $A _{ W}$ mathcalA_W has free rank exactly two. Let $A _{ W} ^{*}$ mathcalA_W^* be the torsion subgroup of $A _{ W}$ mathcalA_W , and $A _{ B} ^{*}$ mathcalA_B^* the corresponding group for the black triangles. We show that $A _{ W} ^{*}$ mathcalA_W^* and $A _{ B} ^{*}$ mathcalA_B^* have the same order, and conjecture that they are isomorphic. For each spherical latin trade $W$, we show there is a unique disjoint mate $B$ such that $( W, B)$ is a connected and separated bitrade. The bitrade $( W, B)$ is associated with a two-colourable planar triangulation and we show that $W$ can be embedded in $A _{ W} ^{*}$ mathcalA_W^* , thereby proving a conjecture due to Cavenagh and Drápal. The proof involves constructing a (0,1) presentation matrix whose permanent and determinant agree up to sign. The Smith normal form of this matrix determines $A _{ W} ^{*}$ mathcalA_W^* , so there is an efficient algorithm to construct the embedding. Contrasting with the spherical case, for each genus $g\geq 1$ we construct a latin trade which is not embeddable in any group and another that is embeddable in a cyclic group.

Détail

Keywords: keywords Latin trade, bitrade, Latin square, abelian group, planar triangulation, Smith normal form, permanent