Minimal orientations of colour critical graphs

Abstract

In 1966 T. Gallai asked whether every criticalk-chromatic graph possesses an orientation having just one directed path of lengthk−1. In this note we show that in general the answer is negative, but also that the answer is affirmative whenk≥5 and the graph has maximal degree at mostk.

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References

  1. [1]

    C. Berge:La Théorie des Graphes et ses Applications, Dunod, Paris, (1958).

    Google Scholar 

  2. [2]

    O. V. Borodin, andA. V. Kostochka: On an Upper Bound of a Graph's Chromatic Number, Depending on the Graph's Degree and Density,J. Comb. Theory B 23, (1977), 247–250.

    Google Scholar 

  3. [3]

    P. A. Catlin: Hajós' Graph Colouring Conjecture: Variations and Counterexamples,J. Comb. Theory (B),26, (1979), 268–274.

    Google Scholar 

  4. [4]

    T. Gallai: On Directed Paths and Circuits,Theory of Graphs (eds. P. Erdős, G. Katona), Academic Press, New York, (1968), 115–118.

    Google Scholar 

  5. [5]

    M. Las Vergnas: Sur les circuits dans les sommes compl'et'ees de graphes orient'es,Cahiers du Centre d'Etudes de Recherche Op'erationelle,15 (1973), 231–244.

    Google Scholar 

  6. [6]

    Toft: 75 Graph Coluring Problems,Graph Colourings (eds. R. Nelson, R. Wilson), Pitman Res. Notes,218, Longman (1990), 9–35.

    Google Scholar 

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Youngs, D.A. Minimal orientations of colour critical graphs. Combinatorica 15, 289–295 (1995). https://doi.org/10.1007/BF01200761

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Mathematics Subject Classification (1991)

  • 05 C 15
  • 05 C 20