Exhaustive generation of k‐critical H ‐free graphs

We describe an algorithm for generating all k-critical H-free graphs, based on a method of Hoang et al. (A graph G is k-critical H-free if G is H-free, k-chromatic, and every H-free proper subgraph of G is (k−1)-colorable). Using this algorithm, we prove that there are only finitely many 4-critical (P7,Ck)-free graphs, for both k=4 and k=5. We also show that there are only finitely many 4-critical (P8,C4)-free graphs. For each of these cases we also give the complete lists of critical graphs and vertex-critical graphs. These results generalize previous work by Hell and Huang, and yield certifying algorithms for the 3-colorability problem in the respective classes. In addition, we prove a number of characterizations for 4-critical H-free graphs when H is disconnected. Moreover, we prove that for every t, the class of 4-critical planar Pt-free graphs is finite. We also determine all 52 4-critical planar P7-free graphs. We also prove that every P11-free graph of girth at least five is 3-colorable, and show that this is best possible by determining the smallest 4-chromatic P12-free graph of girth at least five. Moreover, we show that every P14-free graph of girth at least six and every P17-free graph of girth at least seven is 3-colorable. This strengthens results of Golovach et al.