Choosability of P 5 -free graphs ∗ Petr A. Golovach † Pinar Heggernes † Abstract A graph is k-choosable if it admits a proper coloring of its vertices for every assignment of k (possibly different) allowed colors to choose from for each vertex. It is NP-hard to decide whether a given graph is k-choosable for k ≥ 3, and this problem is considered strictly harder than the k-coloring problem. Only few positive results are known on input graphs with a given structure. Here, we prove that the problem is fixed parameter tractable on P 5 -free graphs when parameterized by k. This graph class contains the well known and widely studied class of cographs. Our result is surprising since the parameterized complexity of k-coloring is still open on P 5 -free graphs. To give a complete picture, we show that the problem remains NP-hard on P 5 -free graphs when k is a part of the input. 1 Introduction Graph coloring is one of the most well known and intensively studied prob- lems in graph theory. The k-Coloring problem asks whether the vertices of an input graph G can be colored with k colors such that no pair of ad- jacent vertices receive the same color (such coloring is also called a proper coloring). This problem is known to be NP-complete even when k ≥ 3 is not a part of the input but a fixed constant. Vizing [19] and Erd˝ os et al. [6] introduced a version of graph coloring called list coloring. In list coloring, a set L(v) of allowed colors is given for each vertex v of the input graph, and we want to decide whether a proper col- oring of the graph exists such that each vertex v receives a color from L(v). If G has a list coloring for every assignment of lists of cardinality k to its ver- tices, then G is said to be k-choosable. Hence the k-Choosability problem * This work is supported by the Research Council of Norway. † Department of Informatics, University of Bergen, N-5020 Bergen, Norway. Emails: {Peter.Golovach|Pinar.Heggernes}@ii.uib.no 1