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. It is the opposite of the complete coloring, which instead requires every color pairing to occur at least once. 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 any path of length > 2, which can be 2-colored but has no harmonious coloring with 2 colors.

Some properties of χ_H(G):

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

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.