T-choosability in graphs Original Research Article

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 vertex-coloring (with positive integers) of G such that c(v)ϵS(v) whenever v ϵ V(G) and ¦c(u) − c(w)¦ ϵ T whenever (u, w)ϵE(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 0ϵT, and C1r log n⩽Tr, s-ch(Kn, n) ⩽ C2r log n for some absolute positive constants c1, c2.