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 mn, 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 \chi^\prime_l(K_{n, n}) = n for natural n. It means that the list chromatic index of the complete bipartite graph K_{n, 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.


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