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. © 1974.
Toft, B. (1974). On critical subgraphs of colour-critical graphs. Discrete Mathematics, 7(3–4), 377–392. https://doi.org/10.1016/0012-365X(74)90045-4