Dinitz conjecture

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 χ 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 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.