Harmonious Colorings: bounds, heuristics and integer-linear formulations
For graph theorists and algorithm designers, this work provides improved bounds and new optimization formulations for harmonious coloring, though the improvements are incremental.
This paper improves an upper bound on the harmonious chromatic number by fixing a proof from prior work, and introduces the first integer-linear programming formulations for the problem along with heuristics, with preliminary tests on random and DIMACS instances.
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.