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.

Harmonious coloring of the complete 7-ary tree with 3 levels using 12 colors. The harmonious chromatic number of this graph is 12. Any fewer colors will result in a color pair appearing on more than one pair of adjacent vertices. Moreover, by Mitchem's Formula, χ_H(T_7,3) = ⌈(3/2)(7+1)⌉ = 12.

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 any path of length>2, which can be 2-colored but has no harmonious coloring with 2 colors.

Some properties of χ_H(G):

1. ${\displaystyle \chi _{H}(T_{k,3})=\left\lceil {\frac {3(k+1)}{2}}\right\rceil }$, 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.