Complexity of Coloring Graphs without Paths and Cycles

Let Pt and C` denote a path on t vertices and a cycle on ` vertices, respectively. In this paper we study the k-coloring problem for (Pt, C`)-free graphs. Maffray and Morel, and Bruce, Hoang and Sawada, have proved that 3-colorability of P5-free graphs has a finite forbidden induced subgraphs characterization, while Hoang, Moore, Recoskie, Sawada, and Vatshelle have shown that k-colorability of P5-free graphs for k ≥ 4 does not. These authors have also shown, aided by a computer search, that 4-colorability of (P5, C5)-free graphs does have a finite forbidden induced subgraph characterization. We prove that for any k, the k-colorability of (P6, C4)-free graphs has a finite forbidden induced subgraph characterization. We provide the full lists of forbidden induced subgraphs for k = 3 and k = 4. As an application, we obtain certifying polynomial time algorithms for 3-coloring and 4-coloring (P6, C4)-free graphs. (Polynomial time algorithms have been previously obtained by Golovach, Paulusma, and Song, but those algorithms are not certifying; in fact they are not efficient in practice, as they depend on multiple use of Ramsey-type results and resulting tree decompositions of very high widths.) To complement these results we show that in most other cases the k-coloring problem for (Pt, C`)free graphs is NP-complete. Specifically, for ` = 5 we show that k-coloring is NP-complete for (Pt, C5)-free graphs when k ≥ 4 and t ≥ 7; for ` ≥ 6 we show that k-coloring is NP-complete for (Pt, C`)-free graphs when k ≥ 5, t ≥ 6; and additionally, for ` = 7, we show that k-coloring is also NP-complete for (Pt, C7)-free graphs if k = 4 and t ≥ 9. This is the first systematic study of the complexity of the k-coloring problem for (Pt, C`)-free graphs. We almost completely classify the complexity for the cases when k ≥ 4, ` ≥ 4, and identify the last three open cases.