An upper bound of radio k-coloring problem and its integer linear programming model

For a positive integer k, a radio k-coloring of a simple connected graph G = (V(G), E(G)) is a mapping \(f:V(G) \to \{ 0,1,2, \ldots \}\) such that \(|f(u) - \,f(v)| \ge k + 1 - d(u, \, v)\) 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, rc_k(f), is the maximum integer assigned by it to some vertex of G. The radio k-chromatic number, rc_k(G) of G is min{rc_k(f)}, where the minimum is taken over all radio k-colorings f of G. If k is the diameter of G, then rc_k(G) is known as the radio number of G. In this paper, we propose an improved upper bound of radio k-chromatic number for a given graph against the other which is due to Saha and Panigrahi (in: Arumugan, Smyth (eds) Combinatorial algorithms (IWOCA 2012). Lecure notes in computer science, vol 7643, Springer, Berlin, 2012). The computational study shows that the proposed algorithm overcomes the previous algorithm. We introduce a polynomial algorithm [differs from the other that is due to Liu and Zhu (SIAM J Discrete Math 19(3):610–621, 2005)] which determines the radio number of the path graph P_n. Finally, we propose a new integer linear programming model for the radio k-coloring problem. The computational study between the proposed algorithm and LINGO solver shows that the proposed algorithm overcomes LINGO solver.