New results on the robust coloring problem

TL;DR

This paper introduces new theoretical results on the robust coloring problem, including complexity proofs, bounds, and properties, providing the first non-heuristic insights into this NP-hard graph coloring variation.

Contribution

It presents the first non-heuristic results for the robust coloring problem, including complexity analysis, bounds, and properties, advancing understanding beyond heuristic approaches.

Findings

Proves robust coloring better approximates equitable partitions.

Shows NP-completeness for the two-color case.

Solves a conjecture on paths related to the problem.

Abstract

Many variations of the classical graph coloring model have been intensively studied due to their multiple applications; scheduling problems and aircraft assignments, for instance, motivate the robust coloring problem. This model gets to capture natural constraints of those optimization problems by combining the information provided by two colorings: a vertex coloring of a graph and the induced edge coloring on a subgraph of its complement; the goal is to minimize, among all proper colorings of the graph for a fixed number of colors, the number of edges in the subgraph with the endpoints of the same color. The study of the robust coloring model has been focused on the search for heuristics due to its NP-hard character when using at least three colors, but little progress has been made in other directions. We present a new approach on the problem obtaining the first collection of non-heuristic results for general graphs; among them, we prove that robust coloring is the model that better approaches the equitable partition of the vertex set, even when the graph does not admit a so-called \emph{equitable coloring}. We also show the NP-completeness of its decision problem for the unsolved case of two colors, obtain bounds on the associated robust coloring parameter, and solve a conjecture on paths that illustrates the complexity of studying this coloring model.