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

A no-hole 2-distant coloring of a graph Γ is an assignment c of nonnegative integers to the vertices of Γ such that | c (v) - c (w) | ≥ 2 for any two adjacent vertices v and w, and the integers used are consecutive. Whenever such a coloring exists, define 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) by using a group-theoretic approach.