The notion of Cayley color graphs of groups is generalized to inverse semigroups and groupoids. The set of partial automorphisms of the Cayley color graph of an inverse semigroup or a groupoid is isomorphic to the original inverse semigroup or groupoid. The groupoid of color permuting partial automorphisms of the Cayley color graph of a transitive groupoid is isomorphic to the original groupoid.

The modified bubble-sort graph of dimension n is the Cayley graph of Sn generated by n cyclically adjacent transpositions. In the present paper, it is shown that the automorphism group of the modified bubble sort graph of dimension n is Sn×D2n, for all n ≥ 5. Thus, a complete structural description of the automorphism group of the modified bubble-sort graph is obtained. A similar direct product...

Let p be a prime and n a positive integer. In [J. Austral. Math. Soc. 81 (2006), 153–164], Feng and Kwak showed that if p > 5 then every connected cubic symmetric graph of order 2p is a Cayley graph. Clearly, this is not true for p = 5 because the Petersen graph is non-Cayley. But they conjectured that this is true for p = 3. This conjecture is confirmed in this paper. Also, for the case when p...

A graph is called a GRR if its automorphism group acts regularly on its vertex-set. Such a graph is necessarily a Cayley graph. Godsil has shown that there are only two infinite families of finite groups that do not admit GRRs: abelian groups and generalised dicyclic groups [4]. Indeed, any Cayley graph on such a group admits specific additional graph automorphisms that depend only on the group...

For a finite group G and a subset S ⊆ G (possibly, S contains the identity of G), the bi-Cayley graph BCay(G, S) of G with respect to S is the graph with vertex set G × {0, 1} and with edge set {(x, 0), (sx, 1)|x ∈ G, s ∈ S}. A bi-Cayley graph BCay(G, S) is called a BCI-graph if, for any bi-Cayley graph BCay(G, T ), whenever BCay(G, S) ∼= BCay(G, T ) we have T = gS , for some g ∈ G, α ∈ Aut(G)....

We construct a 2-generated group Γ such that its Cayley graph possesses finite connected subsets with arbitrarily large finite Heesch number. Thus we obtain an example of a Cayley graph with an infinite Heesch number.

For any finite abelian group G and any subset S ⊆ G, we determine the connectivity of the addition Cayley graph induced by S on G. Moreover, we show that if this graph is not complete, then it possesses a minimum vertex cut of a special, explicitly described form. 1. Background: addition Cayley graphs For a subset S of the abelian group G, we denote by Cay+G(S) the addition Cayley graph induced...

Nathanson was the pioneer in introducing the concepts of Number Theory, particularly, the "Theory of Congruences" in Graph Theory, thus paving way for the emergence of a new class of graphs, namely "Arithmetic Graphs". Cayley graphs are another class of graphs associated with the elements of a group. If this group is associated with some arithmetic function then the Cayley g...

The question of which groups admit planar Cayley graphs goes back over 100 years, being settled for finite groups by Maschke in 1896. Since that time, various authors have studied infinite planar Cayley graphs which satisfy additional special conditions. We consider the question of which groups possess any planar Cayley graphs at all by categorizing such graphs according to their connectivity. ...

The Cayley Isomorphism property for combinatorial objects was introduced by L. Babai in 1977. Since then it has been intensively studied for binary relational structures: graphs, digraphs, colored graphs etc. In this paper we study this property for oriented Cayley maps. A Cayley map is a Cayley graph provided by a cyclic rotation of its connection set. If the underlying graph is connected, the...