Dinitz conjecture explained

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.

equals . That is, if each edge of the complete bipartite graph is assigned a set of colors, it is possible to choose one of the assigned colors for each edgesuch that no two edges incident to the same vertex have the same color.

Galvin's proof generalizes to the statement that, for every bipartite multigraph, the list chromatic index equals its chromatic index. The more general edge list coloring conjecture states that the same holds not only for bipartite graphs, but also for any loopless multigraph. An even more general conjecture states that the list chromatic number of claw-free graphs always equals their chromatic number. The Dinitz theorem is also related to Rota's basis conjecture.