The complexity of planar graph choosability

Abstract

A graph G is k-choosable if for every assignment of a set S(v) of k colors to every vertex v of G, there is a proper coloring of G that assigns to each vertex v a color from S(v). We consider the complexity of deciding whether a given graph is k-choosable for some constant k. In particular, it is shown that deciding whether a given planar graph is 4-choosable is NP-hard, and so is the problem of deciding whether a given planar triangle-free graph is 3-choosable. We also obtain simple constructions of a planar graph which is not 4-choosable and a planar triangle-free graph which is not 3-choosable.

Multiple list colouring of planar graphs

“…A smaller non-4-choosable planar graph was constructed [6], a 3-colourable non-4-choosable planar graph was constructed by Gutner [3,9] and it was prove in [3] that it is NP-complete to decide if a given planar graph is 4-choosable.…”

Choosability on H-free graphs

“…On the positive side, 2-Choosability can be solved in polynomial time on all graphs [7]. The problems 3-Choosability and 4-Choosability remain Π p 2 -complete on planar graphs [15], whereas every planar graph is 5-choosable [27].…”

Choosability of P 5-Free Graphs

“…List coloring has received increasing attention since the beginning of 90's, and there are very good surveys [1,17] and books [11] on the subject. It is proved to be a very difficult problem; Gutner and Tarsi [9] proved that k-Choosability is Π P 2 -complete for bipartite graphs for any fixed k ≥ 3, whereas 2-Choosability can be solved in polynomial time [6]. The 3-Choosability and 4-Choosability problems remain Π P 2 -complete for planar graphs, whereas any planar graph is 5-choosable [16].…”