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On quasiprimitive \(pqr\)-graphs. (English) Zbl 0838.05060

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 \(r\).

MSC:

05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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