Divisibility and T-span of graphs

Abstract

Assume that G is a simple graph and T is a T-set (a finite set of nonnegative integers satisfying 0∈T). A T-coloring of G is a function c that assigns an integer (color) c(v) to each vertex v of G in such a way that if two vertices u,v are adjacent then |c(u)−c(v)| is not in T. The T-span of G, denoted by sp_T(G), is the minimal span over all possible T-colorings of G, where the span of a T-coloring c of G is the distance between the smallest and the largest color used by c. In this paper we study properties of the T-span related to divisibility. In particular, we show that if a positive integer d divides all elements of the set $\text{{0,1,…,}\text{max}\text{T+1}⧹T}$ then sp_T(G) is divisible by d and the quotient sp_T(G)/d equals sp_S(G), where $\text{S={t∈}\text{Z}\text{:}\text{d·t∈T}}$. We also show that for any positive integer d, if $\text{S={0,1,…,d·(}\text{max}\text{T+1)}⧹{d·t:∉T}}$ then sp_S(G)=d·sp_T(G). As a result of these considerations we obtain some simple formulas describing the T-span for many new T-sets and we discover some new families of $\text{E}$ and $\text{G}$, where $\text{E}$ is the collection of such T-sets T that the equality sp_T(G)=sp_T(K_χ(G)) holds for every graph G and $\text{G}$ is the collection of such T-sets T that the greedy (or first-fit) algorithm produces optimal T-colorings for all complete graphs. A full characterization of bipartite graphs in terms of T-colorings and T-span is also given.