Complexity of coloring graphs without paths and cycles

Abstract

Let Pt and Cl denote a path on t vertices and a cycle on l vertices, respectively. In this paper we study the k-COLORING problem for (Pt, Cl)-free graphs. It has been shown by Golovach, Paulusma, and Song that when l = 4 all these problems can be solved in polynomial time. By contrast, we show that in most other cases the k-COLORING problem for (Pt, Cl)-free graphs is NP-complete. Specifically, for l = 5we show that k-COLORING is NP-complete for (P t, C5)-free graphs when k ≥ 4 and t ≥ 7; for l ≥ 6 we show that k-COLORING is NP-complete for (Pt, Cl)-free graphs when k ≥ 5, t ≥ 6; and additionally, we prove that 4-COLORING is NP-complete for (Pt, Cl)-free graphs when t ≥ 7 and l ≥ 6 with l ≠ 7, and that 4-COLORING is NP-complete for (Pt, Cl)- free graphs when t ≥ 9 and l ≥ 6 with l ≠ 9. It is known that, generally speaking, for large k the k-COLORING problem tends to remain NPcomplete when one forbids an induced path Pt with large t. Our findings mean that forbidding an additional induced cycle Cl (with ≥ 4) does not help. We also revisit the problem of k-COLORING (P t, C4)-free graphs, in the case t = 6. (For t = 5 the k-COLORING problem is known to be polynomial even on just P5-free graphs, for every k.) The algorithms of Golovach, Paulusma, and Song are not practical as they depend on Ramsey-type results, and end up using tree-decompositions with very high widths. We develop more practical algorithms for 3-COLORING and 4-COLORING on (P6, C4)-free graphs. Our algorithms run in linear time if a clique cutset decomposition of the input graph is given. Moreover, our algorithms are certifying algorithms. We provide a finite list of all minimal non-k-colorable (P6, C4)-free graphs, for k = 3 and k = 4. Our algorithms output one of these minimal obstructions whenever a kcoloring is not found. In fact, we prove that there are only finitely many minimal non-k-colorable (P6, C4)-free graphs for any fixed k; however, we do not have the explicit lists for higher k, and thus no certifying algorithms. (We note there are infinitely many non-k-colorable P5-free, and hence P6-free, graphs for any given k ≥ 4, according to a result of Hoang, Moore, Recoskie, Sawada, and Vatshelle.) © 2014 Springer-Verlag Berlin Heidelberg.