Regular maps of graphs of order 4<Emphasis Type="Italic">p</Emphasis>

A 2-cell embedding f: X → S of a graph X into a closed orientable surface S can be described combinatorially by a pair \(\mathcal{M} = (X;\rho )\) called a map, where ρ is a product of disjoint cycle permutations each of which is the permutation of the arc set of X initiated at the same vertex following the orientation of S. It is well known that the automorphism group of \(\mathcal{M}\) acts semi-regularly on the arc set of X and if the action is regular then the map \(\mathcal{M}\) and the embedding f are called regular. Let p and q be primes. Du et al. J. Algebraic Combin., 19, 123–141 (2004)]_classified the regular maps of graphs of order pq. In this paper all pairwise non-isomorphic regular maps of graphs of order 4p are constructed explicitly and the genera of such regular maps are computed. As a result, there are twelve sporadic and six infinite families of regular maps of graphs of order 4p; two of the infinite families are regular maps with the complete bipartite graphs K _2p,2p as underlying graphs and the other four infinite families are regular balanced Cayley maps on the groups ?_4p , ? ₂ ² × ? _p and D _4p .