A new lower bound on the number of edges in colour-critical graphs and hypergraphs

Authors:
Alexandr V. Kostochka;Michael Stiebitz
Affiliations:
University of Illinois, Urbana, IL and Novosibirsk State Unitersity, 630090 Novosibirsk, Russia;Technische Universität Ilmenau, D-98684 Ilmenau, Germany
Venue:
Journal of Combinatorial Theory Series B
Year:
2003

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Abstract

A graph G is called k-critical if it has chromatic number k, but every proper subgraph of G is (k- 1)-colourable. We prove that every k-critical graph (k≥ 6) on n ≥ k + 2 vertices has at least 1/2(k- 1 + (k-3)/(k-c)(k-1)+k-3) n edges where c = (k - 5)(1/2- 1/ (k-1)(k-2)). This improves earlier bounds established by Gallai (Acad. Sci. 8 (1963) 165) and by Krivelevich (Combinatorica 17 (1999) 401).