A Lower Bound for the t-Tone Chromatic Number of a Graph in Terms of Wiener Index

Abstract

A t-tone k-coloring of a graph $G=(V,E)$ is a function $f:V\rightarrow {[k] \atopwithdelims ()t}$ such that $|f(u)\cap f(v)|<d(u,v)$ for all distinct vertices u and v. The t-tone chromatic number of G, denoted $\tau _t(G)$, is the smallest positive integer k such that G has a t-tone k-coloring. The Wiener index W(G) of a connected graph G is the sum of the distances of all pairs of vertices of G. In this paper, we prove that $\tau _t(G)\ge t|V|-W(G)+{|V|\atopwithdelims ()2}$ for a connected graph G and obtain a characterization when the equality holds. As a result, for each graph G (not necessarily connected), we obtain a formula for the t-tone chromatic number of G when t is sufficiently large.