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
T. Gallai, Kritische Graphen I,Magyar Tud. Akad. Mat. Kutató Int. Közl. 8 (1963), 165–192.
T. Gallai, Kritische Graphen II.Magyar Tud. Akad. Mat. Kutató Int. Közl. 8 (1963), 373–395.
L. Lovász,Combinatorial Problems and Exercises, Akad. Kiadó, Budapest 1979.
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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DOI: https://doi.org/10.1007/BF02579239