No-hole 2-distant colorings for Cayley graphs on finitely generated abelian groups

A no-hole 2-distant coloring of a graph ΓΓ is an assignment c   of nonnegative integers to the vertices of ΓΓ such that |c(v)-c(w)|⩾2|c(v)-c(w)|⩾2 for any two adjacent vertices vv and ww, and the integers used are consecutive. Whenever such a coloring exists, define nsp(Γ)nsp(Γ) to be the minimum difference (over all c  ) between the largest and smallest integers used. In this paper we study the no-hole 2-distant coloring problem for Cayley graphs over finitely generated abelian groups. We give sufficient conditions for the existence of no-hole 2-distant colorings of such graphs, and obtain upper bounds for the minimum span nsp(Γ)nsp(Γ) by using a group-theoretic approach.