-- As is shown in  any bipartite multigraph G has the following property. If for some regular edge-colouring of G with n colours, n > 3, all the subgraphs generated by the edges of any three colours are uniquely 3-colourable, then G is uniquely n-colourable. In the present paper we show that, for any n > 4 and m > 3, there exist -regular multigraphs with 2m vertices which do not possess the mentioned property. These multigraphs are counter-examples to the Kotzig problem  on transformations of edge-colourings of regular multigraphs. We consider finite non-oriented multigraphs without loops. (Harary's terminology  is used.) Let G be an -regular multigraph with 2m vertices, n > 3. A subset of m pairwise non-adjacent edges of G is called a 1-factor of G. A colouring in which each edge is coloured in one of n colours and adjacent edges are coloured in different colours is called a regular colouring of edges of G with n colours or simply -colouring. For -colouring of G, a subgraph, which is generated by all the edges of some k colours, is called a fc-colour subgraph. A multigraph G is called -colourable if it has
Discrete Mathematics and Applications – de Gruyter
Published: Jan 1, 1993
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