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doi:10.1016/S0012-365X(96)00214-2 
Copyright © 1997 Published by Elsevier Science B.V.
Triangulations of 3-way regular tripartite graphs of degree 4, with applications to orthogonal latin squares
Received 23 June 1995;
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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
, 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.
