Radio numbers of some classes of GP(n, 1) and Cin(1, r)

For any simple connected graph G with diameter d and an integer k, 1 ≤ k ≤ d, a radio k-coloring is an assignment f of positive integers to the vertices of G such that |f(u)-f(v)|≥, 1 + k - d(u, v), where u and v are any two distinct vertices of G and d(u, v) is the distance between u and v. The maximum color (positive integer) assigned by f to some vertex of G is called the span of f. The minimum of spans of all possible radio k-colorings of G is called the radio k-chromatic number of G, denoted by rc_k(G). For k = d, the coloring is called the radio coloring and the radio d-chromatic number is the radio number of G. In this article, we give a criteria for a radio coloring to be minimal. We use this technique to define minimal radio colorings of some classes of generalized Petersen graphs and circulant graphs and find their radio numbers.