We consider the bandwidth coloring problem, a generalization of the well-known graph coloring problem. For the latter problem, a classical theorem, discovered independently by Gallai, Roy and Vitaver, states that the chromatic number of a graph is bounded from above by the number of vertices in the longest elementary path in any directed graph derived by orienting all edges in the graph. We generalize this result to the bandwidth coloring problem. Two proofs are given, a simple one and a more complex that is based on a series of equivalent mathematical programming models. These formulations can motivate the development of various solution algorithms for the bandwidth coloring problem.