A lower bound for radio [formula omitted]-chromatic number

Abstract

For a positive integer k with 1⩽k⩽q, a radio k-coloring of a simple connected graph G=(V(G),E(G)) is a mapping f:V(G)→{0,1,2,…} such that |f(u)−f(v)|⩾k+1−d(u,v) for each pair of distinct vertices u,v∈V(G), where q is the diameter of G and d(u,v) is the distance between u and v. The span rck(f) of f is defined as maxu∈V(G)f(u). The radio k-chromatic number rck(G) of G is min{rck(f)} over all radio k-colorings f of G. In this paper, we give a lower bound of rck(G) for general graph G and discuss the sharpness in the particular case when k=q,q−1. In some cases we give the necessary and sufficient conditions for equality of this bound. As an application we obtain lower bounds of the radio k-chromatic number for the graphs Cn,Pm□Pn,Km□Cn,Pm□Cn and Qn. Moreover, we show that the lower bound of rck(Qn) is an improvement of the existing one.