A conjecture on bipartite graphical regular representations

Abstract

In this paper we are concerned with the classification of the finite groups admitting a bipartite DRR and a bipartite GRR.First, we find a natural obstruction in a finite group for not admitting a bipartite GRR. Then we give a complete classification of the finite groups satisfying this natural obstruction and hence not admitting a bipartite GRR. Based on these results and on some extensive computer computations, we state a conjecture aiming to give a complete classification of the finite groups admitting a bipartite GRR.Next, we prove the existence of bipartite DRRs for most of the finite groups not admitting a bipartite GRR found in this paper. Actually, we prove a much stronger result: we give an asymptotic enumeration of the bipartite DRRs over these groups. Again, based on these results and on some extensive computer computations, we state a conjecture aiming to give a complete classification of the finite groups admitting a bipartite DRR.

On the existence and the enumeration of bipartite regular representations of Cayley graphs over abelian groups

“…However, we observe that the work of Fitzpatrick, Hegarty, Liebeck, and MacHale [8,10,14] on groups admitting automorphisms inverting many elements seems to be relevant. (During the refereeing process of this paper, we have given an explicit description of the groups satisfying the second condition in Conjecture 1.15, see [7]. )…”

On Haar digraphical representations of groups

“…For instance, we classify the finite groups G having an abelian subgroup A of index 2 admitting a bipartite DRR having bipartition {A, G \ A}. The analogous classification for Cayley graphs is much harder, see [9,10]. In fact, we do not have a classification of finite groups admitting a Haar graphical representation and hence we have no classification of finite groups admitting an n-partite graphical representation, when n ≥ 2.…”