Abstract
The notion of list-T-coloring is a common generalization of T-coloring and 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 c(v) of each vertex v is contained in its list L(v) and |c(u)-c(v)| \notin T for any two adjacent vertices u and v.
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 K_2 is finite. For the case when the T-choice number of K_2 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 K_2.