Edge-symmetric Orientable Imbeddings of Complete Graphs

Abstract

In this paper the classification of the edge-symmetric imbeddings of the complete graphs K n in orientable surfaces is completed. Biggs showed that regular imbeddings of complete graphs in oriented surfaces exist iff n is a prime power p ie , his examples being Cayley maps based on the finite field F = GF ( n ). It has since been shown [5] that these are the only examples, and that there are φ ( n - 1)/ e isomorphism classes of such maps (where φ is Euler's function), each corresponding to a conjugacy class of primitive elements of F . We show that non-regular edge-symmetric imbeddings of complete graphs in oriented surfaces exist iff n is a prime power p e greater than 3 and congruent to 3 mod 4, that they too are Cayley maps based on the finite field F = GF ( n ), and that there arc 1/4( n - 3) φ ( n - 1)/ e isomorphism classes of such maps.