Ádám's conjecture is true in the square-free case
The reviewer [J. Comb. Theory 2, 393 (1967); see also Acta Cybern. 3, 187-214 (1977; Zbl 0374.94037)] conjectured that two circulant directed graphs having \(n\) vertices are isomorphic only if a certain special isomorphism (by means of a number \(r\) relatively prime to \(n\)) can be established between them. Partial results and incomplete attempts, due to several authors, are crowned in the present article: a full proof of the conjecture is achieved in the case of a square-free \(n\). Involved algebraic discussions are carried out, using techniques from the theory of permutation groups and the theory of Schur rings. The basic idea is to analyze the 2-closed permutation groups \(G\) fulfilling \(C_n\leq G\leq {\mathfrak S}(C_n)\), where \(C_n\) is the cyclic group of order \(n\) and \(n\) is supposed to be square-free. It turns out that any two regular cyclic subgroups of \(G\) are then conjugate. (This is shown in three steps: (i) \(G\) is primitive, (ii) the number \(k\) of elements of the imprimitivity system is prime, (iii) \(k\) is composite.) Hence, the truth of the conjecture follows from a theorem of \textit{L. Babai} [Acta. Math. Acad. Sci. Hungar. 29, 329-336 (1977; Zbl 0378.05035)]. It is worthy being noticed that the conjecture is known to be false if \(n\) is divisible by 8 or by the square of an odd prime, and it is (in general) undecided in the remaining case (i.e. when \(n/4\) is a square- free odd integer).
scientific article; zbMATH DE number 802352
scientific article