Harmonious coloring explained

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 of a graph is the minimum number of colors needed for any harmonious coloring of .

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

Some properties of :

\chi _{H} (T _{k,3} ) = \left\lceil (3 (k +1 ))/(2) \right\rceil, where is the complete -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

• 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. .
• 10.1016/0012-365X(89)90207-0 . Mitchem . J. . 1989 . On the harmonious chromatic number of a graph . Discrete Math. . 74 . 1–2 . 151–157 . free .