Improving Lower Bounds for Equitable Chromatic Number

TL;DR

This paper presents a new integer linear programming method to improve lower bounds for the equitable chromatic number of graphs, with computational validation on benchmark datasets.

Contribution

A novel two-stage ILP-based approach for better lower bounds on the equitable chromatic number of graphs.

Findings

Improved lower bounds for equitable chromatic number on benchmark graphs.

Effective ILP formulation for equitable coloring problems.

Enhanced understanding of equitable coloring constraints.

Abstract

In many practical applications the underlying graph must be as equitable colored as possible. A coloring is called equitable if the number of vertices colored with each color differs by at most one, and the least number of colors for which a graph has such a coloring is called its equitable chromatic number. We introduce a new integer linear programming approach for studying the equitable coloring number of a graph and show how to use it for improving lower bounds for this number. The two stage method is based on finding or upper bounding the maximum cardinality of an equitable color class in a valid equitable coloring and, then, sequentially improving the lower bound for the equitable coloring number. The computational experiments were carried out on DIMACS graphs and other graphs from the literature.