Harmonious colouring of line graph of commuting and non-commuting graph of D_{2n}

Abstract

Harmonious coloring of graph \(G\) is a proper vertex coloring, where each pair of colors occurs at most on one pair of adjacent vertices. Minimum number of colors required for Harmonious coloring of \(G\) is the harmonious chromatic number, \(\chi_{H}(G)\). Here we determine the Harmonious chromatic number of the line graph of commuting graph and non-commuting graph of the dihedral group, \(D_{2n}\).