A Classification of Regular Embeddings of Graphs of Order a Product of Two Primes

Abstract

In this paper, we classify the regular embeddings of arc-transitive simple graphs of order pq for any two primes p and q (not necessarily distinct) into orientable surfaces. Our classification is obtained by direct analysis of the structure of arc-regular subgroups (with cyclic vertex-stabilizers) of the automorphism groups of such graphs. This work is independent of the classification of primitive permutation groups of degree p or degree pq for p ≠ q and it is also independent of the classification of the arc-transitive graphs of order pq for p ≠ q.