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Harmonious Graph

A connected Labeled Graph with $n$ Edges in which all Vertices can be labeled with distinct Integers (mod $n$) so that the sums of the Pairs of numbers at the ends of each Edge are also distinct (mod $n$). The Ladder Graph, Fan, Wheel Graph, Petersen Graph, Tetrahedral Graph, Dodecahedral Graph, and Icosahedral Graph are all harmonious (Graham and Sloane 1980).

See also Graceful Graph, Labeled Graph, Postage Stamp Problem, Sequential Graph


Gallian, J. A. ``Open Problems in Grid Labeling.'' Amer. Math. Monthly 97, 133-135, 1990.

Gardner, M. Wheels, Life, and other Mathematical Amusements. New York: W. H. Freeman, p. 164, 1983.

Graham, R. L. and Sloane, N. ``On Additive Bases and Harmonious Graphs.'' SIAM J. Algebraic Discrete Math. 1, 382-404, 1980.

Guy, R. K. ``The Corresponding Modular Covering Problem. Harmonious Labelling of Graphs.'' §C13 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 127-128, 1994.

© 1996-9 Eric W. Weisstein