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,[1] and proved in 1994 by Fred Galvin.[2][3]
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 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 edge
such 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.[4] The Dinitz theorem is also related to Rota's basis conjecture.[5]
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In combinatorics, the Dinitz theorem (formerly known as Dinitzconjecture) is a statement about the extension of arrays to partial Latin squares, proposed...
politician Yefim Dinitz, Soviet and Israeli computer scientist The now-proven Dinitzconjecture about partial Latin squares, made by Jeff Dinitz Dinic's algorithm...
proposing the Dinitz conjecture, which became a major theorem. Dinitz was born in 1952 in Brooklyn, New York City, New York. Dinitz is also well known for...
Kakeya sets in vector spaces over finite fields Bregman–Minc inequality Dinitz problem Steve Fisk's proof of the art gallery theorem Five proofs of Turán's...
and was considered to be "great progress" on the Dinitzconjecture. The remaining case of the conjecture for squares (balanced complete bipartite graphs)...
chromatic index is always at least as large as the chromatic index. The Dinitzconjecture on the completion of partial Latin squares may be rephrased as the...
combinatorics. His notable combinatorial work includes the proof of the Dinitzconjecture. In set theory, he proved with András Hajnal that if ℵω1 is a strong...
Combinatorics, CRC Press, p. 212, ISBN 978-1-4398-0623-4 Colbourn, Charles J.; Dinitz, Jeffrey H. (2 November 2006). Handbook of Combinatorial Designs (2nd ed...
125 (1–3): 201–209. doi:10.1016/0012-365X(94)90161-9. Colbourn C. J.; Dinitz J. H. "The CRC Handbook of Combinatorial Designs", CRC, 1996. Grünbaum B...
1007/978-3-540-33783-6_18. ISBN 978-3-540-33782-9. C. J. Colbourn; Jeffrey H. Dinitz (2 November 2006). Handbook of Combinatorial Designs. CRC Press. pp. 525...
example, L. Storme does in his chapter on Finite Geometry in Colbourn & Dinitz (2007, pg. 702) Technically this is a rank two incidence structure, where...
inputs might produce the same hash is often ignored. Colbourn, Charles J.; Dinitz, Jeffrey H. (2007), Handbook of Combinatorial Designs (2nd ed.), Boca Raton:...
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