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# Isomorphic Factorization

Isomorphic factorization colors the edges a given graph with colors so that the colored subgraphs are isomorphic. The graph is then -splittable, with as the divisor, and the subgraph as the factor.

When a complete graph is 2-split, a self-complementary graph results. Similarly, an -regular class 1 graph can be -split into graphs consisting of disconnected edges, making the edge chromatic number.

The complete graph can be 3-split into identical planar graphs.

Some Ramsey numbers have been bounded via isomorphic factorizations. For instance, the complete graph has the Clebsch graph as a factor, proving (Gardner 1989). That is, the complete graph on 16 points can be three-colored so that no triangle of a single color appears. (In 1955, was proven.)

In addition, can be 8-split with the Petersen graph as a factor, or 5-split with a doubled cubical graph as a factor (shown by Exoo in 2005).

The Hoffman-Singleton graph is 7-splittable into edges (shown by Royle in 2004). Whether the Hoffman-Singleton graph is a factor of via another 7-split is an unsolved problem.

Complete Graph, Edge Coloring, Isomorphic Graphs

This entry contributed by Ed Pegg, Jr. (author's link)

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## References

Farrugia, A. "Self-complementary Graphs and Generalizations: A Comprehensive Reference Manual." Masters Thesis. University of Malta, 1999. http://members.lycos.co.uk/afarrugia/sc-graph/sc-graph-survey.ps.Gardner, M. "Ramsey Theory." Ch. 17 in Penrose Tiles and Trapdoor Ciphers... and the Return of Dr. Matrix, reissue ed. New York: W. H. Freeman, pp. 231-247, 1989.West, D. "Hoffman-Singleton Decomposition of ." http://www.math.uiuc.edu/~west/openp/hoffsing.html.

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Isomorphic Factorization

## Cite this as:

Pegg, Ed Jr. "Isomorphic Factorization." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/IsomorphicFactorization.html