On critical subgraphs of colour-critical graphs

Abstract

Some results on the distribution of critical subgraphs in colour-critical graphs are obtained. Characterizations of k-critical graphs in terms of their (k − 1)-critical subgraphs are given. The special case k = 4 is considered, and it is proved that if a 4-critical graph I′ has a vortex x of large valency (compared to the number of vertices of I′ not adjacent to x), then I′ contains vertices of valency 3. Finally, a list of all 4-critical graphs with ≤ 9 vertices is exhibited.