Harmonious coloring

In graph theory, a harmonious coloring is a (proper) vertex coloring in which every pair of colors appears on at most one pair of adjacent vertices. The harmonious chromatic number χ_H(G) of a graph G is the minimum number of colors needed for any harmonious coloring of G.

Every graph has a harmonious coloring, since it suffices to assign every vertex a distinct color; thus χ_H(G) ≤ |V(G)|. There trivially exist graphs G with χ_H(G) > χ(G) (where χ is the chromatic number); one example is the path of length 2, which can be 2-colored but has no harmonious coloring with 2 colors.

Some properties of χ_H(G):

1. χ H ( T k , 3 ) = ⌈ 3 ( k + 1 ) 2 ⌉ , where T_k,3 is the complete k-ary tree with 3 levels. (Mitchem 1989)

Harmonious coloring was first proposed by Harary and Plantholt (1982). Still very little is known about it.