Odd automorphisms

An automorphism (or symmetry) of a combinatorial graph may be called even or odd according to whether it acts as an even or odd permutation on the vertices of the graph. In this talk, I will present a partial result about the following question: When does the existence of even automorphisms of a graph force existence of odd automorphisms? In particular, I will present complete information the on existence of odd automorphisms for cubic symmetric graphs.