Certifying coloring algorithms for graphs without long induced paths

Let P-k be a path, C-k a cycle on k vertices, and K-k,K-k a complete bipartite graph with k vertices on each side of the bipartition. We prove that (1) for any integers k, t > 0 and a graph H there are finitely many subgraph minimal graphs with no induced P-k and K-t,K-t that are not H-colorable and (2) for any integer k > 4 there are finitely many subgraph minimal graphs with no induced P-k that are not Ck-2-colorable. The former generalizes the result of Hell and Huang (2013) and the latter extends a result of Bruce etal. (2009). Both our results lead to polynomial-time certifying algorithms for the corresponding coloring problems. (C) 2018 Elsevier B.V. All rights reserved.