Minimal orientations of colour critical graphs


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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Youngs, D.A. Minimal orientations of colour critical graphs. Combinatorica 15, 289–295 (1995).

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Mathematics Subject Classification (1991)

  • 05 C 15
  • 05 C 20