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 χH(G) of a graph G is the minimum number of colors needed for any harmonious coloring of G.
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 Frank, Harary and Plantholt (1982). Still very little is known about it.
External links
[1] (http://www.maths.dundee.ac.uk/~kedwards/biblio.html) A Bibliography of Harmonious Colourings and Achromatic Number by Keith Edwards
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.