Approximate and exact results for the harmonious chromatic number

TL;DR

This paper investigates the computational complexity of harmonious graph coloring, proving APX-hardness for minimum coloring and providing exact values for specific graph classes, advancing both theoretical understanding and practical algorithms.

Contribution

It establishes the APX-hardness of finding minimum harmonious colorings and determines exact harmonious chromatic numbers for certain classes of graphs, including 3-regular planar graphs.

Findings

Finding minimum harmonious coloring is APX-hard.

The greedy algorithm achieves a () approximation ratio.

Exact harmonious chromatic numbers are determined for specific graph classes.

Abstract

Graph colorings is a fundamental topic in graph theory that require an assignment of labels (or colors) to vertices or edges subject to various constraints. We focus on the harmonious coloring of a graph, which is a proper vertex coloring such that for every two distinct colors i, j at most one pair of adjacent vertices are colored with i and j. This type of coloring is edge-distinguishing and has potential applications in transportation network, computer network, airway network system. The results presented in this paper fall into two categories: in the first part of the paper we are concerned with the computational aspects of finding a minimum harmonious coloring and in the second part we determine the exact value of the harmonious chromatic number for some particular graphs and classes of graphs. More precisely, in the first part we show that finding a minimum harmonious coloring for arbitrary graphs is APX-hard, the natural greedy algorithm is a $\Omega(\sqrt{n})$-approximation, and, moreover, we show a relationship between the vertex cover and the harmonious chromatic number. In the second part we determine the exact value of the harmonious chromatic number for all 3-regular planar graphs of diameter 3, some non-planar regular graphs and cycle-related graphs.