The Edge-Orbit Conjecture of Babai

Abstract

This paper proves the Edge-Orbit Conjecture stated by L. Babai (1981, in "Combinatorics" (H. N. V. Temperley, Ed.), pp. 1-40, Cambridge Univ. Press, London). We say a graph X represents a group G if Aut( X) ≅ G. Let m e ( G) be the minimum number of edge orbits among all graphs K which represent G. The Edge-Orbit Conjecture was that m e ( G) is unbounded when G ranges over all finite groups. We show this is true using p-groups of class two and exponent p. The proof uses a characterization theorem from a recent paper of L. Babai, A. J. Goodman, and L. Lovász (1991, European J. Cambin. 12) to bound m e ( G) from below when all subgroups of G are either "small" enough or "large" enough (so that there will be enough automorphisms leaving any small set of subgroups invariant). Then a probabilistic proof is used to show non-constructively that plenty of groups exist whose subgroups have this property. The proof shows that there is an infinite family of groups G for which m e ( G) has a lower bound proportional to [formula].