NDLI: Latin trades in groups defined on planar triangulations

Content Provider	Springer Nature Link
Author	Cavenagh, Nicholas J. Wanless, Ian M.
Copyright Year	2008
Abstract	For a finite triangulation of the plane with faces properly coloured white and black, let $\mathcal{A}_{W}$ 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 $\mathcal{A}_{W}$ has free rank exactly two. Let $\mathcal{A}_{W}^{}$ be the torsion subgroup of $\mathcal{A}_{W}$ , and $\mathcal{A}_{B}^{}$ the corresponding group for the black triangles. We show that $\mathcal{A}_{W}^{}$ and $\mathcal{A}_{B}^{}$ 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 $\mathcal{A}_{W}^{}$ , 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 $\mathcal{A}_{W}^{}$ , so there is an efficient algorithm to construct the embedding. Contrasting with the spherical case, for each genus g≥1 we construct a latin trade which is not embeddable in any group and another that is embeddable in a cyclic group.We construct a sequence of spherical latin trades which cannot be embedded in any family of abelian groups whose torsion ranks are bounded. Also, we show that any trade that can be embedded in a finitely generated abelian group can be embedded in a finite abelian group. As a corollary, no trade can be embedded in a free abelian group.
Starting Page	323
Ending Page	347
Page Count	25
File Format	PDF
ISSN	09259899
Journal	Journal of Algebraic Combinatorics
Volume Number	30
Issue Number	3
e-ISSN	15729192
Language	English
Publisher	Springer US
Publisher Date	2008-12-18
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Latin trade Bitrade Latin square Abelian group Planar triangulation Smith normal form Permanent Group Theory and Generalizations Computer Science Order, Lattices, Ordered Algebraic Structures Convex and Discrete Geometry Combinatorics
Content Type	Text
Resource Type	Article
Subject	Discrete Mathematics and Combinatorics Algebra and Number Theory

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