On characterizing radio [formula omitted]-coloring problem by path covering problem

Abstract

Let G be a finite simple graph. For an integer k≥1, a radio k-coloring of G is an assignment f of non-negative integers to the vertices of G satisfying the condition ∣f(u)−f(v)∣≥k+1−d(u,v) for any two distinct vertices u, v of G. The span of f is the largest integer assigned to a vertex of G by f and radio k-chromatic number of G, denoted by rck(G), is the minimum span over all radio k-colorings of G. For k=2, the radio k-coloring becomes L(2,1) coloring problem. On the other hand, path covering problem deals with finding minimum number of vertex disjoint paths required to exhaust all the vertices of G. Georges et al. (1994) explored an elegant relation between L(2,1)-coloring problem and path covering problem. As an extension of their work, we characterize the radio k-coloring problem for any k≥2 of a graph G by the path covering problem of Gc, where either G is triangle free or there is a Hamiltonian path in each component of Gc. As an application, for any such graph, if the exact value or an upper bound is known for any rcp(G), p≥2, we can get the exact value or an upper bound of rck(G) for all k≥2. Determination of radio k-chromatic numbers of complete multi-partite graphs, a certain family of circulant graphs and join of circulant graphs of a certain family are among some other applications.