[Paper Review] Dominator Colorings of Digraphs

Michael Cary|arXiv (Cornell University)|Feb 19, 2019

Advanced Graph Theory Research31 references4 citations

TL;DR

This paper introduces dominator colorings for directed graphs (digraphs), establishing the minimum dominator chromatic number across all orientations of paths and cycles. It reveals that subgraphs can have higher dominator chromatic numbers than their supergraphs—contradicting standard coloring intuition—and introduces a new invariant, $\varsigma^\star(D)$, measuring the discrepancy between dominator and chromatic numbers due to orientation.

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.

Motivation & Objective

• To extend the concept of dominator coloring from undirected graphs to directed graphs (digraphs).
• To determine the minimum dominator chromatic number $\chi_d(D)$ over all possible orientations of a given graph structure.
• To investigate whether the dominator chromatic number of a subdigraph can exceed that of its superdigraph, challenging standard coloring monotonicity.
• To define and study a new graph invariant $\varsigma^\star(D)$, measuring the maximum difference between $\chi_d(D)$ and $\chi(G_D)$ across all orientations of the underlying undirected graph.
• To identify open problems and future research directions in dominator colorings of digraphs.

Proposed method

• Define a dominator coloring of a digraph $D$ as a proper vertex coloring where every vertex dominates some color class via outgoing arcs.
• Use $\chi_d(D)$ to denote the minimum number of color classes in any dominator coloring over all orientations of the underlying undirected graph $G_D$.
• Analyze directed paths $P_n$ and directed cycles $C_n$ by characterizing their out-degree sequences and identifying optimal colorings.
• Prove that $\chi_d(P_4) = 3 > 2 = \chi_d(C_4)$, demonstrating that subgraphs can have larger dominator chromatic numbers than supergraphs.
• Introduce the invariant $\varsigma(D) = \chi_d(D) - \chi(G_D)$ and its maximum variant $\varsigma^\star(D) = \max\{\varsigma(D)\}$ over all orientations of $G_D$, to quantify orientation-induced deviation from standard chromatic number.
• Establish closed-form expressions for $\varsigma^\star(\cdot)$ on paths and cycles based on $n \mod 4$.

Experimental results

Research questions

• RQ1 Which digraphs satisfy $\chi_d(D) = \chi(D)$, i.e., for which graphs does $\varsigma^\star(D) = 0$?
• RQ2 When is $\chi_d(D)$ invariant under all orientations of the underlying graph $G_D$?
• RQ3 When is $\chi_d(D)$ invariant under arc reversal, i.e., $\chi_d(D) = \chi_d(D^{-})$?
• RQ4 Can Theorem 3, which characterizes graphs with $\chi_d(D) = 2$, be generalized to larger domination structures in partite sets?
• RQ5 What families of digraphs satisfy $\limsup_{n\to\infty} \frac{\chi_d(D)}{\Delta(D)} = r$ for some $r \in \mathbb{R}$?

Key findings

• The dominator chromatic number of a digraph is not monotonic with respect to subgraphs: $\chi_d(P_4) = 3 > 2 = \chi_d(C_4)$, showing that a subgraph can have a larger dominator chromatic number than its supergraph.
• For directed paths, $\varsigma^\star(P_n)$ is given by $3k-2$ if $n=4k$, $3k-1$ if $n=4k+1$ or $n=4k+2$, and $3k$ if $n=4k+3$. For directed cycles, $\varsigma^\star(C_n)$ is $3k-2$ for $n=4k$ or $n=4k+1$, $3k-1$ for $n=4k+2$, and $3k$ for $n=4k+3$.
• The invariant $\varsigma^\star(D)$ is always non-negative, as $\chi_d(D) \geq \chi(G_D)$ by Observation 1.
• For complete graphs $K_n$ and complete bipartite graphs $K_{m,n}$ with all arcs directed from one part to the other, $\varsigma^\star(K_n) = \varsigma^\star(K_{m,n}) = 0$, meaning their dominator chromatic number matches their chromatic number.
• The dominator chromatic number of a digraph is not bounded by its maximum degree, indicating a fundamental difference from standard vertex coloring.
• The paper identifies that $\chi_d(D)$ can increase under orientation, and introduces $\delta(D,H) = \chi_d(H) - \chi_d(D)$ to quantify such discrepancies in subgraph relationships.