combinatorial Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ap ...

mathematics, Toida's conjecture, due to

Shunichi Toida Shun'ichi or Shunichi (written: or ) is a masculine Japanese given name. Notable people with the name include: *, Japanese baseball player and manager *, Japanese academic *, Japanese footballer *, Japanese sprint canoeist *, Japanese engineer *, ...

in 1977, is a refinement of the disproven

Ádám's conjecture In graph theory, a circulant graph is an undirected graph acted on by a cyclic group of symmetries which takes any vertex to any other vertex. It is sometimes called a cyclic graph, but this term has other meanings. Equivalent definitions Ci ...

from 1967.

Statement

Both conjectures concern

circulant graph In graph theory, a circulant graph is an undirected graph acted on by a cyclic group of symmetries which takes any vertex to any other vertex. It is sometimes called a cyclic graph, but this term has other meanings. Equivalent definitions Cir ...

s. These are graphs defined from a positive integer

n

and a set

S

of positive integers. Their vertices can be identified with the numbers from 0 to

n-1

, and two vertices

i

and

j

are connected by an edge whenever their difference modulo

n

belongs to set

S

. Every symmetry of the

cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bi ...

of addition modulo

n

gives rise to a symmetry of the

n

-vertex circulant graphs, and Ádám conjectured (incorrectly) that these are the only symmetries of the circulant graphs. However, the known counterexamples to Ádám's conjecture involve sets

S

in which some elements share non-trivial divisors with

n

. Toida's conjecture states that, when every member of

S

relatively prime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equival ...

n

, then the only symmetries of the circulant graph for

n

and

S

are symmetries coming from the underlying cyclic group.

Proofs

The conjecture was proven in the special case where ''n'' is a prime power by Klin and Poschel in 1978, and by Golfand, Najmark, and Poschel in 1984. The conjecture was then fully proven by Muzychuk, Klin, and Poschel in 2001 by using

Schur algebra In mathematics, Schur algebras, named after Issai Schur, are certain finite-dimensional algebras closely associated with Schur–Weyl duality between general linear and symmetric groups. They are used to relate the representation theories ...

, and simultaneously by Dobson and

Morris Morris may refer to: Places Australia *St Morris, South Australia, place in South Australia Canada * Morris Township, Ontario, now part of the municipality of Morris-Turnberry * Rural Municipality of Morris, Manitoba ** Morris, Manit ...

in 2002 by using the

classification of finite simple groups In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or els ...

Notes

Combinatorics Conjectures that have been proved {{combin-stub