Independent sets ink-chromatic graphs


A graph is said to have propertyP k if in eachk-colouring ofG using allk colours there arek independent vertices having all colours. An (unpublished) suggestion of P. Erdős is answered in the affirmative: For eachk≧3 there is a k-critical graph withP k . With the aid of a construction of T. Gallaik-chromatic graphs (k≧7) withP k orP k+1 of arbitrarily high connectivity are obtained. The main result is: Eachk-chromatic graph (k≧3) of girth ≧6 hasP k or is a circuit of length 7.

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Dedicated to Paul Erdős on his seventieth birthday

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Voss, H.J. Independent sets ink-chromatic graphs. Combinatorica 5, 261–269 (1985).

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AMS subject classification (1980)

  • 05 C 15