We describe an algorithm for generating all k-critical H-free
graphs, based on a method of Ho\`{a}ng et al. 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 graphs (P8,C4)-free graphs. For each case 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.
Moreover, we prove that for every t, the class of 4-critical planar
Pt-free graphs is finite. We also determine all 27 4-critical planar
(P7,C6)-free graphs.
We also prove that every P10-free graph of girth at least five is
3-colorable, and determine the smallest 4-chromatic P12-free graph of
girth five. Moreover, we show that every P13-free graph of girth at least
six and every P16-free graph of girth at least seven is 3-colorable. This
strengthens results of Golovach et al.Comment: 17 pages, improved girth results. arXiv admin note: text overlap with
arXiv:1504.0697