Colour-critical graphs with few edges Original Research Article

A graph G is called k-critical if G is k-chromatic but every proper subgraph of G has chromatic number at most k − 1. In this paper the following result is proved. If G is a k-critical graph (k ⩾ 4) on n vertices, then 2|E(G)| > (k − 1)n + ((k − 3)(k2 − 3))n + k − 4 where n ⩾ k + 2 and n ≠ 2k − 1. This improves earlier bounds established by Dirac (1957) and Gallai (1963).