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 χH(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 χH(G) ≤ |V(G)|. There trivially exist graphs G with χH(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 χH(G):
- χH(Tk,3) = ⌈(3/2)(k+1)⌉, where Tk,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.
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—Bible: Hebrew Psalm CXLVII (Paraphrased by The Countess of Pembroke)