Harmonious Chromatic Number of At Least One Degree of Vertices in Some Graphs

A harmonious coloring is a proper vertex coloring such that every pair of colors occurs not more than single edge, the smallest quantity of colors in harmonious coloring is a harmonious chromatic number and it is denoted by χH(G). This paper shows that the harmonious chromatic number of at least one degree of vertices in certain graphs like the Hurdle graph Hdn, Bull graph B(G) and the complement of the
Bull graph. Each graph of the vertices has degree at least one, i.e., d(vi) ≥1, for every vertex v ∈ V(G) of degree at least 1, the neighbors of v receive distinct colors and also different color pair of vertices.