Dominator Colorings of Digraphs

Abstract

This paper serves as the first extension of the topic of dominator colorings of graphs to the setting of digraphs. We establish the dominator chromatic number over all possible orientations of paths and cycles. In this endeavor we discover that there are infinitely many counterexamples of a graph and subgraph pair for which the subgraph has a larger dominator chromatic number than the larger graph into which it embeds. Finally, a new graph invariant measuring the difference between the dominator chromatic number of a graph and the chromatic number of that graph is established and studied. The paper concludes with some of the possible avenues for extending this line of research.

A Linear Algorithm for Minimum Dominator Colorings of Orientations of Paths

“…The algorithm works by sequentially through the vertex set, from v 1 through v n , and colors each vertex in such a a way as to minimize the number of colors used in a proper dominator coloring. From Theorem 1 in [4] we know that all vertices with in-degree equal to zero must belong to the same color class. For this reason, the first thing the algorithm checks for is precisely this.…”

“…, 0, 2, 0} (it is possible that one or both of the end vertices of such a subpath do not have out-degree zero, but since all of the out-degree two vertices are assigned the same color, and since such an end vertex would have out-degree two in the full path, we may ignore end vertices of these subpaths that do not have out-degree zero). From Theorem 1 in [4] we know that such oriented paths are minimized in terms of dominator colorings precisely when the first vertex is assigned a color C ⋆ which is used for all vertices of out-degree zero that are not uniquely colored, and when this color is assigned to every other vertex with out-degree zero (every fourth vertex in the path). For convenience, we refer to these paths as 2-chains since they are maximal subpaths with respect to the density of vertices with out-degree two.…”

“…As is stands, the algorithm would minimally color the reversal of this particular orientation of P 6 but not this particular orientation of P 6 . Because this was the only exception listed in Theorem 1 from [4], and because checking an input (n) will not alter the time complexity of the algorithm, we simply check for this occurrence in the same if statement as when we check to see if a 2-chain of length three exists.…”

“…The first results on dominator colorings of digraphs were the dominator chromatic number of paths and cycles [4], and that the dominator chromatic number of digraphs is that the dominator chromatic number of a tree is invariant under reversal of orientation [3]. Other interesting results established in [4] on dominator colorings of digraphs include the fact that the dominator chromatic number of a subgraph H ⊂ G can be larger than the dominator chromatic number of G, as well as that lim sup…”

Dominator Chromatic Numbers of Orientations of Trees

“…These results led to the vast development of applications of dominating sets in undirected networks, including, e.g., [3,9,[14][15][16]. Recently the notion of dominator colorings was extended to directed graphs in [4]. In that paper the focus was on finding the dominator chromatic number over all possible orientations of paths and cycles.…”