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Ádám’s conjecture is true in the square-free case. (English) Zbl 0833.05063
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 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).

##### MSC:
 05C60 Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.) 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures 20C05 Group rings of finite groups and their modules (group-theoretic aspects)
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