Embeddings of symmetric graphs in surfaces

The context of this paper is the study of <::~n'TIrnpl~rv Every object, concrete or abstract, real or imaginary, has a certain amount of symmetry which is measured by a group. Since the days of Cayley [1), a group has been a mathematical object obeying certain a'doms and the question arises: given an abstract group, of what is it a group of ""71rnrnpj'.rH'''' A very answer, described in more detail in Section 1, is: groups tend to act as groups of automorphisms of symmetric Such a graph is formed, for example, the vertices and edges of a cube (or any of the regular solids), and the Euclidean group of motions of the cube acts on it as a group of graph au tomorphisms. Alternatively, this graph can be formed from the surface of a cube by removing the interiors of its 6 faces. The opposite procedure will be investigated here: given a graph on which a group acts symmetrically, how can some of its cycles be filled in to form a surface on which the action of the members of the group extend to isometries or homeomorphisms? Even for the graph of a cube the answer is not obvious: certainly, six of its 4-cycles can be filled in to give the ordinary cube, but it is also possible to fill in the four 6-cycles obtained by leaving out pairs of opposite vertices and their incident edges, to get a torus. In other words, the graph