Complexity of Coloring Graphs without Paths and Cycles

Abstract

Abstract. 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.

A Survey on the Computational Complexity of Coloring Graphs with Forbidden Subgraphs

“… proved that 4‐ Precoloring Extension is ‐complete for ‐free graphs, which implies (ii):4, and they also proved (ii):7. We observe that (ii):2, (ii):3, and (ii):5 follow immediately from corresponding results for k ‐ Coloring as shown by Hell and Huang . Finally, we consider k ‐ Coloring ; first the ‐complete cases.…”

“… proved that the Grötzsch graph is the only 4‐critical ‐free graph. Hell and Huang showed that, for all , the number of k ‐vertex‐critical ‐free graphs is finite. Moreover, they gave an explicit construction of all four 4‐vertex‐critical ‐free graphs and of all 13 5‐vertex‐critical ‐free graphs.…”

Coloring graphs without short cycles and long induced paths

“…Very recently, Hell and Huang [16] extended the results in Table 2 by showing that k-COLORING is NP-complete on (C r , P )-free graphs in the following cases:…”

“…The aforementioned results of Hell and Huang [16], which improve the results in Table 2, combined with our result for k = 4, r = 3 and = 164 and the known polynomial-time results for k-COLORING on P -free graphs (namely the cases k ≥ 1, ≤ 5 [17] and k = 3, = 6 [31]) leave a number of cases open in the computational complexity classification of k-COLORING for (C r , P )-free graphs. An intriguing question is to determine the computational complexity of 3-COLORING restricted to (C 3 , P )-free graphs for all integers .…”

Exhaustive generation of k‐critical ‐free graphs

“… have shown that the only 4‐critical ‐free graph is the Grötzsch graph. More recently, Hell and Huang proved that there are four 4‐critical ‐free graphs. They also proved that in general, there are only finitely many k ‐critical ‐free graphs.…”