Recognizing and Testing Isomorphism of Cayley Graphs over an Abelian Group of Order 4p in Polynomial Time

On Cayley representations of central Cayley graphs over almost simple groups

“…First, we assume that a Cayley graph Γ = Cay(G, X) is central, i.e., the connection set X of Γ is normal: X g = X for every g ∈ G. Obviously, every Cayley graph over an abelian group is central, it explains why central graphs are also called quasiabelian in [21,23]. The problem of finding pairwise nonequivalent Cayley representations was solved for cyclic groups [6] and for some abelian groups of small rank [17,19]. We will show that such results are still possible even for groups that are very far from abelian, if we restrict ourselves to central Cayley representations that is Cayley representations of central Cayley graphs.…”

The Cayley isomorphism property for the group $C_4\times C_p^2$

“…The Cayley representation problem of graphs (see [20,27,28]) which seems to be very hard in general becomes easier in case when the input group is a DCI-group. Recall that a Cayley representation of a graph Γ over a group G is defined to be an isomorphism from Γ to a Cayley graph over G. Two Cayley representations of Γ are called equivalent if the images of Γ under these representations are Cayley isomorphic.…”

“…The Cayley representation problem can be formulated as follows: given a group G and a graph Γ find a full set of non-equivalent Cayley representations of Γ over G. From the definition of a DCI-group it follows that every graph has at most one Cayley representation over a DCI-group up to equivalence. More information on the Cayley representation problem and its connection with DCI-groups can be found in [20,27].…”