In combinatorial mathematics, Toida's conjecture, due to Shunichi Toida in 1977,[1] is a refinement of the disproven Ádám's conjecture from 1967.
Both conjectures concern circulant graphs.
Every symmetry of the cyclic group of addition modulo
However, the known counterexamples to Ádám's conjecture involve sets
Toida's conjecture states that, when every member of
are symmetries coming from the underlying cyclic group.
The conjecture was proven in the special case where n is a prime power by Klin and Poschel in 1978,[2] and by Golfand, Najmark, and Poschel in 1984.
[3] The conjecture was then fully proven by Muzychuk, Klin, and Poschel in 2001 by using Schur algebra,[4] and simultaneously by Dobson and Morris in 2002 by using the classification of finite simple groups.
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