Choosability of graphs with infinite sets of forbidden differences Original Research Article

The notion of the list-T-coloring is a common generalization of the T-coloring and the list-coloring. Given a set of non-negative integers T, a graph G and a list-assignment L, the graph G is said to be T-colorable from the list-assignment L if there exists a coloring c such that the color image of each vertex image is contained in its list image and image for any two adjacent vertices u and image. The T-choice number of a graph G is the minimum integer k such that G is T-colorable for any list-assignment L which assigns each vertex of G a list of at least k colors. We focus on list-T-colorings with infinite sets T. In particular, we show that for any fixed set T of integers, all graphs have finite T-choice number if and only if the T-choice number of image is finite. For the case when the T-choice number of image is finite, two upper bounds on the T-choice number of a graph G are provided: one being polynomial in the maximum degree of the graph G, and the other being polynomial in the T-choice number of image.