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Colouring Cubic Graphs by Small Steiner Triple Systems

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Abstract

Given a Steiner triple system , we say that a cubic graph G is -colourable if its edges can be coloured by points of in such way that the colours of any three edges meeting at a vertex form a triple of . We prove that there is Steiner triple system of order 21 which is universal in the sense that every simple cubic graph is -colourable. This improves the result of Grannell et al. [J. Graph Theory 46 (2004), 15–24] who found a similar system of order 381. On the other hand, it is known that any universal Steiner triple system must have order at least 13, and it has been conjectured that this bound is sharp (Holroyd and Škoviera [J. Combin. Theory Ser. B 91 (2004), 57–66]).

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References

  1. Archdeacon D. S., Coverings of graphs by cycles, Congr. Numer. 53, 7–14 (1986)

    Google Scholar 

  2. Archdeacon D. S., Generalizations of Tait colouring cubic graphs, Problems in Topological Graph Theory, http://www.emba.uvm.edu/~archdeac/problems/petecol.htm

  3. Biggs N. L., White A. T., Permutation Groups and Combinatorial Structures, Cambridge Univ. Press, Cambridge, 1979

  4. Colbourn C. J., Rosa A., Triple Systems, Oxford Univ. Press, Oxford, 1999

  5. Doyen J., Wilson R. M., Embeddings of Steiner triple systems, Discrete Math. 5, 229–239 (1973)

    Google Scholar 

  6. Duffin R. J., Topology of series-parallel networks, J. Math. Analysis and Appl. 10 (1965), 303–318

    Google Scholar 

  7. Grannell M. J., Griggs T. S., Knor M., Škoviera M., A Steiner triple system which colors all cubic graphs, J. Graph Theory 46, 15–24 (2004)

    Google Scholar 

  8. Holroyd F. C., Škoviera M., Colouring of cubic graphs by Steiner triple systems, J. Combin. Theory Ser. B 91, 57–66 (2004)

    Google Scholar 

  9. Holyer I., NP-completness of edge-colouring, SIAM J. Comp. 10, 718–720 (1981)

    Google Scholar 

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Authors and Affiliations

  1. School of Computer Science, University of Waterloo, 200 University Avenue West, Waterloo, N2L 3G1, Ontario, Canada

    Dávid Pál

  2. Department of Computer Science, Faculty of Mathematics, Physics and Informatics, Comenius University, 842 48, Bratislava, Slovakia

    Martin Škoviera

Authors
  1. Dávid Pál
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  2. Martin Škoviera
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Correspondence to Martin Škoviera.

Additional information

Research partially supported by APVT, project 51-027604.

Research partially supported by VEGA, grant 1/3022/06.

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Pál, D., Škoviera, M. Colouring Cubic Graphs by Small Steiner Triple Systems. Graphs and Combinatorics 23, 217–228 (2007). https://doi.org/10.1007/s00373-007-0696-1

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  • DOI: https://doi.org/10.1007/s00373-007-0696-1

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