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. It can also be formulated as a result in graph theory, that the list chromatic index of the complete bipartite graph ${\displaystyle K_{n,n}}$ equals ${\displaystyle n}$. That is, if each edge of the complete bipartite graph is assigned a set of ${\displaystyle n}$ colors, it is possible to choose one of the assigned colors for each edge such that no two edges incident to the same vertex have the same color.

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