Factor d-domatic colorings of graphs

Consider a graph and a collection of (not necessarily edge-disjoint) connected spanning subgraphs (factors) of the graph. We consider the problem of coloring the vertices of the graph so that each color class of the vertices dominates each factor. We find upper and lower bounds on α(t, k), which we define as the minimum radius of domination d such that every graph with a collection of k factors can be vertex colored with t colors so that each color class d-dominates each factor. It is perhaps surprising that the upper bound is finite and does not depend on the order of the graph. We obtain similar results for a variant of the problem where the number of colors is equal to the number of factors and each color class must d-dominate only the corresponding factor rather than all factors.