On Graph Fall-Coloring: Existence and Constructions

Abstract

Graph fall-coloring, also known as idomatic partition or independent domatic partition of graphs, was formally introduced by Dunbar, Hedetniemi, Hedetniemi, Jacobs, Knisely, Laskar and Rall in 2000 as an extension of both Graph Coloring and Graph Domination. It asks for a partition of the vertex set of a given graph into independent dominating sets, or equivalently into maximal independent sets. We study two fundamental questions related to this concept: when such a partition of vertices can exist, and how it relates to a proper coloring. We construct graphs with a large number of possible fall-colorings and as a consequence of that we answer a question of Dunbar et al. (J Combin Math Combin Comput 33:257–273, 2000) by constructing a family of graphs with arbitrarily far apart chromatic number and fall-chromatic number. In fact, we construct graphs whose Fall set (collection of k such that the graph has a fall coloring with k colors) is an arbitrarily long arithmetic sequence, thus giving us graphs with Fall set of large order having large gaps between its elements. We give a sufficient condition on the minimum degree for fall-colorable graphs and characterize the sharpness of this bound. Related to this, we also construct families of graphs which have both a k-fall-coloring and a k-coloring that is not a fall coloring for every k, illustrating the complex relationship between idomatic and non-domatic independent partitions.