Regular orientable imbeddings of complete graphs (Q1070238)

An imbedding of the complete graph on n vertices is called regular if the automorphism group of the imbedding acts transitively on the set of directed edges. N. L. Biggs has shown that a regular imbedding exists if and only if n is a prime power, and gives examples of such imbeddings based on the Galois field of order n. The authors show here that these examples are the only regular imbeddings. They also determine when two such maps are isomorphic and for which n they are reflexible or self- dual. The techniques used are algebraical. The case \(n=25\) is worked out in detail as a nice example.