Upper bound for radio [formula omitted]-chromatic number of graphs in connection with partition of vertex set

Abstract

For a simple connected graph G=(V(G),E(G)) and a positive integer k with 1⩽k⩽diam(G), a radio k-coloring of G is a mapping f:V(G)→{0,1,2,…} such that |f(u)−f(v)|⩾k+1−d(u,v) holds for each pair of distinct vertices u and v of G, where diam(G) is the diameter of G and d(u,v) is the distance between u and v in G. The span of a radio k-coloring f is the number maxu∈V(G)f(u). The main aim of the radio k-coloring problem is to determine the minimum span of a radio k-coloring of G, denoted by rck(G). Due to NP-hardness of this problem, people are finding lower bounds for rck(.) for particular families of graphs or general graphs G. In this article, we concentrate on finding upper bounds of radio k-chromatic number for general graphs and in consequence a coloring scheme depending on a partition of the vertex set V(G). We apply these bounds to cycle graph Cn and hypercube Qn and show that it is sharp when k is close to the diameter of these graphs.