The Structure of 2-Pyramidal 2-Factorizations

A 2-factorization of a simple graph is called -pyramidal if it admits an automorphism group fixing two vertices and acting sharply transitively on the others. Here we show that such a -factorization may exist only if is a cocktail party graph, i.e., with being a -factor. It will be said of the first or second type according to whether the involutions of form a unique conjugacy class or not. As far as we are aware, -factorizations of the second type are completely new. We will prove, in particular, that admits a 2-pyramidal 2-factorization of the second type if and only if (mod 8).