On the Number of Edges in Colour-Critical Graphs and Hypergraphs

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is called k-critical if it has chromatic number k, but every proper sub(hyper)graph of it is (k-1)-colourable. We prove that for sufficiently large k, every k-critical triangle-free graph on n vertices has at least (k-o(k))n edges. Furthermore, we show that every (k+1)-critical hypergraph on n vertices and without graph edges has at least edges. Both bounds differ from the best possible bounds by o(kn) even for graphs or hypergraphs of arbitrary girth.

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Received June 25, 1998

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Kostochka, A., Stiebitz, M. On the Number of Edges in Colour-Critical Graphs and Hypergraphs. Combinatorica 20, 521–530 (2000). https://doi.org/10.1007/s004930070005

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  • AMS Subject Classification (1991) Classes:  05C15