- Dinitz conjecture
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In combinatorics, the Dinitz conjecture is a statement about the extension of arrays to partial Latin squares, proposed in 1979 by Jeff Dinitz, and proved in 1994 by Fred Galvin.
The Dinitz conjecture, now a theorem, is that given an n × n square array, a set of m symbols with m ≥ n, and for each cell of the array an n-element set drawn from the pool of m symbols, it is possible to choose a way of labeling each cell with one of those elements in such a way that no row or column repeats a symbol.
The Dinitz conjecture is closely related to graph theory, in which it can be succinctly stated as
for natural n. It means that the list chromatic index of the complete bipartite graph Kn,n equals n. In fact, Fred Galvin proved the Dinitz conjecture as a special case of his theorem stating that the list chromatic index of any bipartite multigraph is equal to its chromatic index. Moreover, it is also a special case of the edge list coloring conjecture saying that the same holds not only for bipartite graphs, but also for any loopless multigraph.References
- F. Galvin (1995). "The list chromatic index of a bipartite multigraph". Journal of Combinatorial Theory. Series B 63 (1): 153–158. doi:10.1006/jctb.1995.1011.
- Zeilberger, D. (1996). "The Method of Undetermined Generalization and Specialization Illustrated with Fred Galvin's Amazing Proof of the Dinitz Conjecture". The American mathematical monthly 103 (3): 233–239. arXiv:math/9506215.
- Chow, T. Y. (1995). "On the Dinitz conjecture and related conjectures". Discrete Math 145: 145–173. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.72.7005. Retrieved 2009-04-15.
External links
- Weisstein, Eric W., "Dinitz Problem" from MathWorld.
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