Complexity of Coloring Graphs without Paths and Cycles

Abstract

Let P _t and C _ℓ denote a path on t vertices and a cycle on ℓ vertices, respectively. In this paper we study the k-COLORING problem for (P _t ,C _ℓ)-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 (P _t ,C _ℓ)-free graphs is NP-complete. Specifically, for ℓ = 5 we show that k-COLORING is NP-complete for (P _t ,C ₅)-free graphs when k ≥ 4 and t ≥ 7; for ℓ ≥ 6 we show that k-COLORING is NP-complete for (P _t ,C _ℓ)-free graphs when k ≥ 5, t ≥ 6; and additionally, we prove that 4-COLORING is NP-complete for (P _t ,C _ℓ)-free graphs when t ≥ 7 and ℓ ≥ 6 with ℓ ≠ 7, and that 4-COLORING is NP-complete for (P _t ,C _ℓ)-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 NP-complete when one forbids an induced path P _t with large t. Our findings mean that forbidding an additional induced cycle C _ℓ (with ℓ > 4) does not help. We also revisit the problem of k-COLORING (P _t ,C ₄)-free graphs, in the case t = 6. (For t = 5 the k-COLORING problem is known to be polynomial even on just P ₅-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 (P ₆,C ₄)-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 (P ₆,C ₄)-free graphs, for k = 3 and k = 4. Our algorithms output one of these minimal obstructions whenever a k-coloring is not found. In fact, we prove that there are only finitely many minimal non-k-colorable (P ₆,C ₄)-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 P ₅-free, and hence P ₆-free, graphs for any given k ≥ 4, according to a result of Hoàng, Moore, Recoskie, Sawada, and Vatshelle.)