Deciding k-Colorability of P 5-Free Graphs in Polynomial Time

Abstract

The problem of computing the chromatic number of a P 5 -free graph (a graph which contains no path on 5 vertices as an induced subgraph) is known to be NP-hard. However, we show that for every fixed integer k, there exists a polynomial-time algorithm determining whether or not a P 5 -free graph admits a k-coloring, and finding one, if it does.

Three Complexity Results on Coloring P k -Free Graphs

“…Recent results of Hoàng et al [6] imply that this problem is polynomially solvable on P 5 -free graphs. Their algorithm for -Coloring for any fixed is in fact a list-coloring algorithm where the lists are from the set {1, 2, .…”

“…This problem was known to be polynomially solvable for P 5 -free graphs [6] and NP-complete for P 8 -free graphs [16], so there remains one open case.…”

“…Its proof follows from the fact that the decision problem in this case can be modeled and solved as a 2SAT-problem. This approach has been introduced by Edwards [4] and is folklore now, see also [6] and [12].…”

“…Instead of repeating what has been written in so many papers over the years, we also refer to these surveys for motivation and background. Here we continue the study of -Coloring and its variants for P k -free graphs, a problem that has been studied in several earlier papers by different groups of researchers (see, e.g., [2,3,6,[10][11][12]16]). We summarize all these results in the table in Section 5.…”

A Note for the Two Incremental Satisfiability Problem

“…Dvorá k [13] found a linear-time algorithm to decide whether a triangle-free graph in a general surface Σ is 3-colorable. 3-colorability is also polynomially solvable for graphs containing no induced path on 5 vertices [15].…”