# Colouring Square-Free Graphs without Long Induced Paths

The complexity of Colouring is fully understood for H-free graphs, but there are still major complexity gaps if two induced subgraphs H_1 and H_2 are forbidden. Let H_1 be the s-vertex cycle C_s and H_2 be the t-vertex path P_t. We show that Colouring is polynomial-time solvable for s=4 and t≤ 6, strengthening several known results. Our main approach is to initiate a study into the boundedness of the clique-width of atoms (graphs with no clique cutset) of a hereditary graph class. We first show that the classifications of boundedness of clique-width of H-free graphs and H-free atoms coincide. We then show that this is not the case if two graphs are forbidden: we prove that (C_4,P_6)-free atoms have clique-width at most 18. Our key proof ingredients are a divide-and-conquer approach for bounding the clique-width of a subclass of C_4-free graphs and the construction of a new bound on the clique-width for (general) graphs in terms of the clique-width of recursively defined subgraphs induced by homogeneous pairs and triples of sets. As a complementary result we prove that Colouring is -complete for s=4 and t≥ 9, which is the first hardness result on Colouring for (C_4,P_t)-free graphs. Combining our new results with known results leads to an almost complete dichotomy for restricted to (C_s,P_t)-free graphs.

READ FULL TEXT