A new graph radio k-coloring algorithm

Due to the rapid growth in the use of wireless communication services and the corresponding scarcity and the high cost of radio spectrum bandwidth, Channel assignment problem (CAP) is becoming highly important. Radio k-coloring of graphs is a variation of CAP. For a positive integer k, a radio k-coloring of a simple connected graph G = (V, E) is a mapping f from the vertex set V (G) to the set {0, 1, 2, ...} of non-negative integers such that vertical bar f(u) - f(v)vertical bar >= k + 1 - d(u, v) for each pair of distinct vertices u and v of G, where d(u, v) is the distance between u and v in G. The span of a radio k-coloring f, denoted by span(f), is defined as max(v epsilon V(G)) f(v) and the radio k-chromatic number of G, denoted by rc(k)(G), is min(f) {span(f)} where the minimum is taken over all radio k-coloring of G. In this paper, we present two radio k-coloring algorithms for general graphs which will produce radio k-colorings with their spans. For an n-vertex simple connected graph the time complexity of the both algorithm is of O(n(4)). Implementing these algorithms we get the exact value of rc(k)(G) for several graphs (for example, C-n, C-n x P-2, C-4 x C-n, some circulant graph etc.) and many values of k, especially for k = diam(G).