In [76], a number of edge colorings were described that gave rise to various vertex colorings of interest. In one instance, the color of a vertex was defined as the set of colors of the edges incident with the vertex, with the goal to minimize the number of colors so that the resulting coloring is vertex-distinguishing. In this chapter, we consider edge colorings that result in the same vertex coloring, but with a different goal in mind. Here, for a fixed number k of edge colors 1, 2, …, k, we wish to determine graphs of minimum order having the property that for every subset S of [k] = { 1, 2, …, k}, there is a vertex whose color is S.