Marušič, Dragan; Scapellato, Raffaele; Zgrablić, Boris On quasiprimitive \(pqr\)-graphs. (English) Zbl 0838.05060 Algebra Colloq. 2, No. 4, 295-314 (1995). A non-Cayley number is the order of a vertex-transitive graph which is not a Cayley graph. The problem of determining when a given \(n\) is non-Cayley has been reduced to the case where \(n\) is square-free and divisible by three or more primes. (See B. D. McKay and C. E. Praeger, Vertex-transitive graphs which are not Cayley graphs. II., J. Graph Theory, to appear.) In the present paper, a first step towards a classification in the case where \(n\) is a product of three distinct primes is given by the following theorem: Let \(X\) be an imprimitive \(pqr\)-graph, where \(p\), \(q\), and \(r\) are distinct primes; then either: (i) \(X\) is a genuinely imprimitive graph, or (ii) \(X\) is a quasiprimitive graph arising as a generalized orbital graph associated with a triple from Tables A or B (given in the paper). Moreover, triples from Table A give rise to imprimitive graphs having blocks of composite length \(pq\), whereas triples from Table B give rise to imprimitive graphs having blocks of prime length \(r\). Reviewer: A.T.White (Kalamazoo) Cited in 11 Documents MSC: 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) Keywords:vertex-transitive graph; Cayley graph; imprimitive graph; quasiprimitive graph; orbital graph PDFBibTeX XMLCite \textit{D. Marušič} et al., Algebra Colloq. 2, No. 4, 295--314 (1995; Zbl 0838.05060)