List-coloring apex-minor-free graphs

A graph H is t-apex if H − X is planar for some set X ⊂ V (H) of size t. For any integer t ≥ 0 and a fixed t-apex graph H, we give a polynomial-time algorithm to decide whether a (t + 3)-connected Hminor-free graph is colorable from a given assignment of lists of size t+4. The connectivity requirement is the best possible in the sense that for every t ≥ 1, there exists a t-apex graph H such that testing (t+4)colorability of (t+ 2)-connected H-minor-free graphs is NP-complete. Similarly, the size of the lists cannot be decreased (unless P = NP), since for every t ≥ 1, testing (t+3)-list-colorability of (t+3)-connected Kt+4-minor-free graphs is NP-complete. All graphs considered in this paper are finite and simple. Let G be a graph. A function L which assigns a set of colors to each vertex of G is called a list assignment. An L-coloring φ of G is a function such that φ(v) ∈ L(v) for each v ∈ V (G) and such that φ(u) 6= φ(v) for each edge uv ∈ E(G). For an integer k, we say that L is a k-list assignment if |L(v)| = k for every v ∈ V (G), and L is a (≥k)-list assignment if |L(v)| ≥ k for every v ∈ V (G). The concept of list coloring was introduced by Vizing [27] and Erdős et al. [5]. Clearly, list coloring generalizes ordinary proper coloring; a graph has chromatic number at most k if and only if it can be L-colored for the k-list assignment which assigns the same list to each vertex. Consequently, the computational problem of deciding whether a graph can be colored from a given k-list assignment is NP-complete for every k ≥ 3 [6]. Let this problem be denoted by k-LC. Let us remark that 2-LC is polynomial-time