Ádám's conjecture is true in the square-free case

Ádám's conjecture [1] formulates necessary and sufficient conditions for cyclic (circulant) graphs to be isomorphic. It is known to be true if the number n of vertices is either prime ([4]), a product of two primes ([12]) or satisfies the condition n, φ(n)) = 1, where φ is Euler's function ([15]). On the other hand, it is also known that the conjecture fails if n is divisible by 8 or by an odd square. It was newly conjectured in [15] that Ádám's conjecture is true for all other values of n. We prove that the conjecture is valid whenever n is a square-free number.