An improved lower bound for the radio kk-chromatic number of the hypercube QnQn

For any simple connected graph GG with diameter dd and an integer kk, 1≤k≤d1≤k≤d, a radio  kk-coloring   is an assignment ff of positive integers to the vertices of GG such that |f(u)−f(v)|≥1+k−d(u,v)|f(u)−f(v)|≥1+k−d(u,v), where uu and vv are any two distinct vertices of GG and d(u,v)d(u,v) is the distance between uu and vv. The maximum color (positive integer) assigned by ff to some vertex of GG is called the span   of ff. The minimum of spans of all possible radio kk-colorings of GG is called the radio  kk-chromatic number   of GG, denoted by rck(G). For k=dk=d, the coloring is called the radio coloring   and the radio dd-chromatic number is the radio number   of GG. Kchikech et al. (2008) [3] have given upper and lower bounds for the radio kk-chromatic number of the hypercube QnQn for n≥2n≥2. In this paper, we give an improved lower bound and prove that the bound is sharp for radio coloring.