Proof of a conjecture of T. Gallai concerning connectivity properties of colour-critical graphs

Abstract

Tibor Gallai made the following conjecture. LetG be ak-chromatic colour-critical graph. LetL denote the set of those vertices ofG having valencyk−1 and letH be the rest ofV(G). Then the number of components induced byL is not less than the number of components induced byH, providedL ≠ 0.

We prove this conjecture in a slightly generalized form.

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References

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    H. Sachs andM. Stiebitz, Construction of colour-critical graphs with given major-vertex subgraph,Proc. Int. Coll. Graph Theory and Combinatorics, Marseille-Luminy 1981,to appear.

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Dedicated to Tibor Gallai on his seventieth birthday

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Stiebitz, M. Proof of a conjecture of T. Gallai concerning connectivity properties of colour-critical graphs. Combinatorica 2, 315–323 (1982). https://doi.org/10.1007/BF02579239

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AMS subject classification (1980)

  • 05 C 15
  • 05 C 40