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.

翻译：设$c$为简单图$G$的正常染色，若对于任意两条不同边$uv,xy\in E(G)$，均有$\{c(u),c(v)\}\neq \{c(x),c(y)\}$，则称$c$为和谐染色。图$G$的和谐色数$h(G)$定义为使$G$存在$k$种颜色的和谐染色之最小正整数$k$。本文推广了[Kolay等人在《和谐染色：参数化算法与上界》(Theor. Comp. Sci. 772 (2019), 132-142)]中提出的思想，用于比较图$G$与其通过距离至少为3的顶点识别所得图$H$的和谐色数。此外，通过修正该文献中的某个证明，我们改进了一个上界。我们还首次（据我们所知）提出了该问题的整数线性规划公式及其启发式算法，并给出了随机实例与第二届DIMACS实现挑战赛实例的初步测试结果。