Constructing regular maps and graphs from planar quotients

Let M be a map on an orientable surface. The generic regular map for M is, up to isomorphism, the unique regular map M# such tha t M# covers M and every regular map tha t covers M covers also M#. In this paper, we show tha t several interesting results concerning maps on surfaces and graphs can be established by constructing generic maps over appropriate quotients. Among them are simple proofs of theorems o f V i n c e , M a c B e a t h , and generalizations of results of B r o w n and C o n n e l l y , A r c h d e a c o n , and others. Using the same method we also show tha t for every integer g > 3 there exists an arctransitive cubic graph whose girth equals g.