Harmonious Colorings: bounds, heuristics and integer-linear formulations

TL;DR

This paper investigates harmonious graph colorings, introduces new bounds, compares chromatic numbers of related graphs, and proposes the first integer-linear programming formulations with heuristics, supported by preliminary computational tests.

Contribution

It extends existing methods to compare harmonious chromatic numbers, improves an upper bound, and introduces novel integer-linear formulations and heuristics for the problem.

Findings

Compared harmonious chromatic numbers of related graphs.

Improved an existing upper bound for harmonious coloring.

Developed and tested new integer-linear programming formulations and heuristics.

Abstract

A proper coloring $c$ of a simple graph $G$ is harmonious if, for every pair of distinct edges $uv,xy\in E(G)$, we have that $\{c(u),c(v)\}\neq \{c(x),c(y)\}$. The harmonious chromatic number of $G$, denoted by $h(G)$, is the least positive integer $k$ such that $G$ has a harmonious coloring with $k$ colors. In this work, we extend an idea presented in [Kolay, et al. Harmonious coloring: Parameterized algorithms and upper bounds. Theor. Comp. Sci. 772 (2019), 132-142] to compare the harmonious chromatic numbers of two graphs $G$ and $H$, with $H$ being obtained from $G$ by identifying vertices at distance at least three. Furthermore, by fixing a proof presented in the same work, we manage to improve one of its upper bounds. We also introduce and study the first, to the best of our knowledge, integer-linear programming formulations for this problem in the literature, along with some heuristics. We provide some preliminary tests on random instances and instances from the second DIMACS Implementation Challenge.