Harmonious coloring

File:Harmonious coloring tree.svg

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, Template:Math.

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 Template:Math of a graph Template:Mvar is the minimum number of colors needed for any harmonious coloring of Template:Mvar.

Every graph has a harmonious coloring, since it suffices to assign every vertex a distinct color; thus Template:Math. There trivially exist graphs Template:Mvar with Template:Math (where Template:Math is the chromatic number); one example is any path of Template:Nowrap, which can be 2-colored but has no harmonious coloring with 2 colors.

Some properties of Template:Math:

<math>\chi_{H}(T_{k,3}) = \left\lceil\frac{3(k+1)}{2}\right\rceil,</math>

where Template:Math is the complete [[Glossary of graph theory#k-ary|Template:Mvar-ary]] tree with 3 levels. (Mitchem 1989)

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

See also

• Complete coloring
• Harmonious labeling

External links

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

References

• Template:Cite journal
• Jensen, Tommy R.; Toft, Bjarne (1995). Graph coloring problems. New York: Wiley-Interscience. Template:ISBN.
• Template:Cite journal