T-choosability in graphs

Given a set of nonnegative integers T. and a function ℒ which assigns a set of integers S(v) to each vertex v of a graph G, an ℒ-list T-coloring c of G is a vertex-coloring (with positive integers) of G such that c(v) ∈ S(v) whenever v ∈ V(G) and |c(u) - c(w)| ∉ T whenever (u,w) ∈ E(G). For a fixed T, the T-choice number T-ch(G) of a graph G is the smallest number A such that G has an ℒ-list T-coloring for every collection of sets S(v) of size k each. Exact values and bounds on the T_r,s-choice numbers where T_r,s = {0,s,2s,...,rs} are presented for even cycles, notably that T_r,s-ch(C_2n) = 2r + 2 if n ≥ r + 1. More bounds are obtained by applying algebraic and probabilistic techniques, such as that T-ch(C_2n)≤2|T| if 0 ∈ T, and c₁r log n ≤ T_r,s-ch(K_n,n) ≤ c₂r log n for some absolute positive constants c₁,c₂.