The structure of 2-pyramidal 2-factorizations

A 2-factorization of a simple graph $\Gamma$ is called 2-pyramidal if it admits an automorphism group G fixing two vertices and acting sharply transitively on the others. Here we show that such a 2-factorization may exist only if $\Gamma$ is a cocktail party graph, i.e., $\Gamma = K_2n − I$ with I being a 1-factor. It will be said of the first or second type according to whether the involutions of G form a unique conjugacy class or not. As far as we are aware, 2-factorizations of the second type are completely new. We will prove, in particular, that $K_2n − I$ admits a 2-pyramidal 2-factorization of the second type if and only if n ≡ 1 (mod 8).