Abstract
We give necessary and sufficient conditions for a regular bipartite multigraph of degree m to have an edge-colouring with m colours in which two specified edges receive the same colour. Call a cubic bipartite multigraph G with bipartition (A, B) skew if there is a cutset K of six edges whose removal separates G into two subgraphs G 1 and G 2 such that each edge of K joins a vertex of A ∩ V (G 1 ) to a vertex of B ∩ V (G 2 ). We consider the problem of finding necessary and sufficient conditions for a skew cubic bipartite multigraph to have an edgecolouring with three colours in which three specified edges of K receive different colours. This problem arose in an investigation concerning latin squares. We conclude with a conjecture on this problem.
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