Improved Complexity Results on k-Coloring P t -Free Graphs

Abstract

A graph is H-free if it does not contain an induced subgraph isomorphic to H. We denote by P k and C k the path and the cycle on k vertices, respectively. In this paper, we prove that 4-COLORING is NP-complete for P7-free graphs, and that 5-COLORING is NP-complete for P6-free graphs. These two results improve all previous results on kcoloring Pt-free graphs, and almost complete the classification of complexity of k-COLORING Pt-free graphs for k ≥ 4 and t ≥ 1, leaving as the only missing case 4-COLORING P6-free graphs. We expect that 4-COLORING is polynomial time solvable for P6-free graphs; in support of this, we describe a polynomial time algorithm for 4-COLORING P6free graphs which are also P -free, where P is the graph obtained from C4 by adding a new vertex and making it adjacent to exactly one vertex on the C4.We continue the study of k-COLORING problem for P t -free graphs. This problem has been given wide attention in recent years and much progress has been made through substantial efforts by different groups of researchers [2,3,4,7,11,13,16,17,18,21,24]. We summarize these results and explain our new results below.We refer to [1] for standard graph theory terminology and [9] for terminology on computational complexity. Let G = (V, E) be a graph and H be a set of ⋆ An extended abstract of this paper has appeared in the proceedings of MFCS 2013.

Complexity of Coloring Graphs without Paths and Cycles

“…Recently, Huang [18] proved that the 5-coloring problem for P 6 -free graphs is NP-complete, and that the 4-coloring problem for P 7 -free graphs is also NP-complete. The proof used the following framework.…”

“…We give a reduction from 3-SAT, as in [18]. Let I be any 3-SAT instance with variables X = {x 1 , x 2 , .…”

“…In [18] Huang conjectured that 4-coloring is polynomial time solvable for P 6 -free graphs. If the problems in Problem 1 for k = 4 or Problem 2 are polynomial, this would add evidence to the conjecture.…”

“…Later on, these results were improved by various groups of researchers [3,4,12,21]. The strongest results so far are due to Huang [18] who has proved that 4-coloring is NP-complete for P 7 -free graphs, and that 5-coloring is NP-complete for P 6 -free graphs. On the positive side, Hoàng, Kamiński, Lozin, Sawada, and Shu [15] have shown that k-coloring can be solved in polynomial time on P 5 -free graphs for any fixed k. These results give a complete classification of the complexity of k-coloring P t -free graphs for any fixed k ≥ 5, and leave only 4-coloring P 6 -free graphs open for k = 4.…”

List coloring in the absence of two subgraphs

“…[24] completely determined the computational complexity of Coloring for graph classes characterized by one forbidden induced subgraph. By combining a number of known results, Golovach, Paulusma and Song [15] obtained similar dichotomy results for the problems List Coloring and k-List Coloring, whereas the complexity classifications of the problems List k-Coloring and k-Coloring are still open (for a survey we refer to the paper of Golovach, Paulusma and Song [16] and for some new results to a recent paper of Huang [20]). The following theorem gives these three complexity dichotomies.…”