Triangulations of 3-way regular tripartite graphs of degree 4, with applications to orthogonal latin squares

Abstract

If G is a regular tripartite graph of degree d( G) with tripartition ( A, B, C) of V( G) such that the bipartite subgraphs induced by each of A ∪ B, B ∪ C, C ∪ A are all regular of degree 1 2 d(G) , then we call G 3- way regular. We give necessary and sufficient conditions for a 3-way regular tripartite graph of degree 4 to have a decomposition into edge-disjoint triangles. These yield necessary and sufficient conditions for the completion of a partial latin square of order n in which each row and column is missing exactly two symbols, and in which each symbol occurs exactly n − 2 times. We also give necessary and sufficient conditions for a 3-way regular tripartite graph of degree 4 to have a decomposition into two edge-disjoint parallel classes, each parallel class consisting of disjoint triangles. This in turn yields necessary and sufficient conditions for the completion of a pair of ( n − 2) × n partial orthogonal latin squares. Generalizations of some of the various conditions are shown to be necessary in some more general contexts.