# Counter-examples to the Kotzig problem

Counter-examples to the Kotzig problem -- As is shown in [1] 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 [2] on transformations of edge-colourings of regular multigraphs. We consider finite non-oriented multigraphs without loops. (Harary's terminology [3] 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 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Discrete Mathematics and Applications de Gruyter

# Counter-examples to the Kotzig problem

Discrete Mathematics and Applications, Volume 3 (1) – Jan 1, 1993
4 pages

/lp/de-gruyter/counter-examples-to-the-kotzig-problem-yXzuVGwROS
Publisher
de Gruyter
ISSN
0924-9265
eISSN
1569-3929
DOI
10.1515/dma.1993.3.1.59
Publisher site
See Article on Publisher Site

### Abstract

-- As is shown in [1] 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 [2] on transformations of edge-colourings of regular multigraphs. We consider finite non-oriented multigraphs without loops. (Harary's terminology [3] 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

### Journal

Discrete Mathematics and Applicationsde Gruyter

Published: Jan 1, 1993

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