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. 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); To complement these results we show
that in most other cases the k-coloring problem for (Pt,Cℓ)-free
graphs is NP-complete. Specifically, for ℓ=5 we show that k-coloring is
NP-complete for (Pt,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, for ℓ=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,Cℓ)-free graphs. We almost completely
classify the complexity for the cases when k≥4,ℓ≥4, and
identify the last three open cases