Certifying coloring algorithms for graphs without long induced paths

Let Pk be a path, Ck a cycle on k vertices, and Kk,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 Pk and Kt,t that are not Hcolorable and (2) for any integer k > 4 there are finitely many subgraph minimal graphs with no induced Pk that are not Ck−2-colorable. The former generalizes the result of Hell and Huang [Complexity of coloring graphs without paths and cycles, Discrete Appl. Math. 216: 211– 232 (2017)] and the latter extends a result of Bruce, Hoàng, and Sawada [A certifying algorithm for 3-colorability of P5-Free Graphs, ISAAC 2009: 594–604]. Both our results lead to polynomial-time certifying algorithms for the corresponding coloring problems.