A lower bound for radio kk-chromatic number

For a positive integer kk with 1⩽k⩽q1⩽k⩽q, a radio kk-coloring of a simple connected graph G=(V(G),E(G))G=(V(G),E(G)) is a mapping  f:V(G)→{0,1,2,…}f:V(G)→{0,1,2,…} such that |f(u)−f(v)|⩾k+1−d(u,v)|f(u)−f(v)|⩾k+1−d(u,v) for each pair of distinct vertices u,v∈V(G)u,v∈V(G), where qq is the diameter of GG and d(u,v)d(u,v) is the distance between uu and vv. The span rck(f)rck(f) of ff is defined as maxu∈V(G)f(u)maxu∈V(G)f(u). The radio kk-chromatic number rck(G)rck(G) of GG is min{rck(f)}min{rck(f)} over all radio kk-colorings ff of GG. In this paper, we give a lower bound of rck(G)rck(G) for general graph GG and discuss the sharpness in the particular case when k=q,q−1k=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 kk-chromatic number for the graphs Cn,Pm□Pn,Km□Cn,Pm□CnCn,Pm□Pn,Km□Cn,Pm□Cn and QnQn. Moreover, we show that the lower bound of rck(Qn)rck(Qn) is an improvement of the existing one.