In combinatorics, the Dinitz Theorem (formerly known as 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 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 theorem is closely related to graph theory, in which it can be succinctly stated as for natural . It means that the list chromatic index of the complete bipartite graph equals . In fact, Fred Galvin proved the Dinitz theorem 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.
- 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: .
- Chow, T. Y. (1995). "On the Dinitz conjecture and related conjectures". Discrete Math. 145: 145–173. Retrieved 2009-04-15.
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