Harmonious coloring

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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 harmonius chromatic number χ_H(G) of a graph G is the minimum number of colors needed for any harmonious coloring of G.

Some properties of χ_H(G):

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

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

See also: Complete coloring

[edit] External links

• [1] A Bibliography of Harmonious Colourings and Achromatic Number by Keith Edwards

[edit] References

• Frank, O.; Harary, F.; Plantholt, M. (1982). The line-distinguishing chromatic number of a graph. Ars Combin. 14, 241–252.
• Jensen, Tommy R.; Toft, Bjarne (1995). Graph coloring problems. New York: Wiley-Interscience. ISBN 0-471-02865-7.
• Mitchem, J. (1989). On the harmonious chromatic number of a graph. Discrete Math. 74, 151–157.