Approximate and exact results for the harmonious chromatic number

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