T-choosability in Graphs

Given a set of nonnegative integers T , and a function S which assigns a set of integers S(v) to each vertex v of a graph G, an S-list T -coloring c of G is a vertexcoloring (with positive integers) of G such that c(v) ∈ S(v) whenever v ∈ V (G) and |c(u)− c(w)| 6∈ 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 k such that G has an S-list T -coloring for every collection of sets S(v) of size k each. Exact values and bounds on the Tr,s-choice numbers where Tr,s = {0, s, 2s, . . . , rs} are presented for even cycles, notably that Tr,s-ch(C2n) = 2r + 2 if n ≥ r + 1. More bounds are obtained by applying algebraic and probabilistic techniques, such as that T -ch(C2n) ≤ 2|T | if 0 ∈ T , and c1r log n ≤ Tr,s-ch(Kn,n) ≤ c2r log n for some absolute positive constants c1, c2.