Robert Jajcay

Cayley maps

We present a theory of Cayley maps, i.e., embeddings of Cayley graphs into oriented surfaces having the same cyclic rotation of generators around each vertex. These maps have often been used to encode symmetric embeddings of graphs. We also present an algebraic theory of Cayley maps and we apply the theory to determine exactly which regular or edge-transitive tilings of the sphere or plane are Cayley maps or Cayley graphs. Our main goal, however, is to provide the general theory so as to make it easier for others to study Cayley maps.

Skew-morphisms of regular Cayley maps

A Cayley map M is a 2-cell embedding of a Cayley graph in an orientable surface with the same orientation (the induced permutation of generators) at each vertex. The concept of a skew-morphism generalizes several concepts previously studied with respect to regular Cayley maps, and allows for a unified theory of regular Cayley maps and their automorphism groups. Using algebraic properties of skew-morphisms of groups we reprove or extend some previously known results and obtain several new ones.

Automorphism groups of Cayley maps

Cayley maps are embeddings of Cayley graphs in orientable surfaces, with the same local orientation at every vertex. In this paper, conditions will be given under which a Cayley map is regular or reflexible, together with algorithms for checking these conditions. We shall give the description of the automorphism group of an arbitrary Cayley map and focus on the special case of k-balanced Cayley maps.

Regular t-balanced Cayley maps

The concept of a t-balanced Cayley map is a natural generalization of the previously studied notions of balanced and anti-balanced Cayley maps (the terms coined by [J. Siran, M. Skoviera, Groups with sign structure and their antiautomorphisms, Discrete Math. 108 (1992) 189-202. [12]]). We develop a general theory of t-balanced Cayley maps based on the use of skew-morphisms of groups [R. Jajcay, J. Siran, Skew-morphisms of regular Cayley maps, Discrete Math. 244 (1-3) (2002) 167-179], and apply our results to the specific case of regular Cayley maps of abelian groups.

Constructions of Self-complementary Circulants with No Multiplicative Isomorphisms

The main topic of the paper is the question of the existence of self-complementary Cayley graphs Cay(G, S) with the property S??=G#\S for all ??Aut(G). We answer this question in the positive by constructing an infinite family of self-complementary circulants with this property. Moreover, we obtain a complete classification of primes p for which there exist self-complementary circulants of order p2with this property.

The Structure of Automorphism Groups of Cayley Graphs and Maps

The automorphism groups Aut(C(G, X)) and Aut(CM(G, X, p)) of a Cayley graph C(G, X) and a Cayley map CM(G, X, p) both contain an isomorphic copy of the underlying group G acting via left translations. In our paper, we show that both automorphism groups are rotary extensions of the group G by the stabilizer subgroup of the vertex 1G. We use this description to derive necessary and sufficient conditions to be satisfied by a finite group in order to be the (full) automorphism group of a Cayley graph or map and classify all the finite groups that can be represented as the (full) automorphism group of some Cayley graph or map.

On the girth of voltage graph lifts

We study the potential and limitations of the voltage graph construction for producing small regular graphs of large girth. We determine the relation between the girth of the base graph and the lift, and we show that any base graph can be lifted to a graph of arbitrarily large girth. We determine upper bounds on the girths of voltage graphs with respect to the nilpotency class in the case of nilpotent groups or the length of the derived series in the case of solvable voltage groups. These results suggest the use of perfect groups, which we use to construct the smallest known cubic graphs of girths 29 and 30. We also construct the smallest known (5, 10)-graphs and (7, 8)-graphs.

On the limitations of the use of solvable groups in Cayley graph cage constructions

A (k,g)-cage is a (connected) k-regular graph of girth g having smallest possible order. While many of the best known constructions of small k-regular graphs of girth g are known to be Cayley graphs, there appears to be no general theory of the relationship between the girth of a Cayley graph and the structure of the underlying group. We attempt to fill this gap by focusing on the girth of Cayley graphs of nilpotent and solvable groups, and present a series of results supporting the intuitive notion that the closer a group is to being abelian, the less suitable it is for constructing Cayley graphs of large girth. Specifically, we establish the existence of upper bounds on the girth of Cayley graphs with respect to the nilpotency class and/or the derived length of the underlying group, when this group is nilpotent or solvable, respectively.

Small vertex-transitive graphs of given degree and girth

We investigate the basic interplay between the small k-valent vertex-transitive graphs of girth g and the (k;g)-cages, the smallest k-valent graphs of girth g. We prove the existence of k-valent Cayley graphs of girth g for every pair of parameters k 2 and g 3, improve the lower bounds on the order of the smallest (k;g) vertex-transitive graphs for certain families with prime power girth, and generalize the construction of Bray, Parker and Rowley that has yielded several of the smallest known (k;g)-graphs.

Regular t -balanced Cayley maps for abelian groups

The inherent high symmetry of Cayley maps makes them an excellent source of orientably regular maps, and the regularity of a Cayley map has been shown to be equivalent to the existence of a skew-morphism of its underlying group that has a generating orbit closed under inverses. We set to investigate the properties of the so-called t -balanced skew-morphisms of abelian groups with the aim of providing the basis for a complete classification of t -balanced regular Cayley maps of abelian groups. In the case of cyclic groups, we show that the only t -balanced regular Cayley maps for the groups Z 2 r , Z 2 p r and Z 4 p r , p an odd prime, r ? 1 , are the well understood balanced and antibalanced Cayley maps.