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Minimal and minimum size latin bitrades of each genus

James Lefevre;  Diane Donovan;  Nicholas J. Cavenagh;  Aleš Drápal — 2007

Commentationes Mathematicae Universitatis Carolinae

Suppose that T ∘ and T ☆ are partial latin squares of order n , with the property that each row and each column of T ∘ contains the same set of entries as the corresponding row or column of T ☆ . In addition, suppose that each cell in T ∘ contains an entry if and only if the corresponding cell in T ☆ contains an entry, and these entries (if they exist) are different. Then the pair T = ( T ∘ , T ☆ ) forms a . The of T is the total number of filled cells in T ∘ (equivalently T ☆ ). The latin bitrade is if there is no latin bitrade...

Distinct equilateral triangle dissections of convex regions

Diane M. Donovan;  James G. Lefevre;  Thomas A. McCourt;  Nicholas J. Cavenagh — 2012

Commentationes Mathematicae Universitatis Carolinae

We define a proper triangulation to be a dissection of an integer sided equilateral triangle into smaller, integer sided equilateral triangles such that no point is the vertex of more than three of the smaller triangles. In this paper we establish necessary and sufficient conditions for a proper triangulation of a convex region to exist. Moreover we establish precisely when at least two such equilateral triangle dissections exist. We also provide necessary and sufficient conditions for some convex...

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