Parallelism

EnProblems involving the idea of parallelism occur in finite geometry and in graph theory. This article addresses the question of constructing parallelisms with some degree of "symmetry". In particular, can we say anything on parallelisms admitting an automorphism group acting doubly transitively on "parallel classes"

On the existence spectrum for sharply transitive G-designs, G a [k]-matching

In this paper we consider decompositions of the complete graph Kv into matchings of uniform cardinality k. They can only exist when k is an admissible value, that is a divisor of v(v−1)/2 with 1≤k≤v/2. The decompositions are required to admit an automorphism group Γ acting sharply transitively on the set of vertices. Here Γ is assumed to be either non-cyclic abelian or dihedral and we obtain necessary conditions for the existence of the decomposition when k is an admissible value with 1<k<v/2. Differently from the case where Γ is a cyclic group, these conditions do exclude existence in specific cases. On the other hand we produce several constructions for a wide range of admissible values, in particular for every admissible value of k when v is odd and Γ is an arbitrary group of odd order possessing a subgroup of order gcd(k,v)

Primitive collineation groups of ovals with a fixed point

AbstractWe investigate collineation groups of a finite projective plane of odd order n fixing an oval and having two orbits on it, one of which is assumed to be primitive. The situation in which the group fixes a point off the oval is considered. We prove that it occurs in a Desarguesian plane if and only if (n+1)/2 is an odd prime, the group lying in the normalizer of a Singer cycle of PGL(2,n) in this case. For an arbitrary plane we show that the group cannot contain Baer involutions and derive a number of structural and numerical properties in the case where the group has even order. The existence question for a non-Desarguesian example is addressed but remains unanswered, although such an example cannot have order n≤23 as computer searches carried out with GAP show

A class of complete arcs in multiply derived planes

We prove that unital-derived (q^2 - q + 1)-arcs of PG(2, q^2) still yield complete arcs after multiple derivation with respect to disjoint derivation sets on a given line

Even cycles and even 2-factors in the line graph of a simple graph

Let G be a connected graph with an even number of edges. We show that if the subgraph of G induced by the vertices of odd degree has a perfect matching, then the line graph of G has a 2-factor whose connected components are cycles of even length (an even 2-factor). For a cubic graphG, we also give a necessary and sufficient condition so that the corresponding line graph L(G) has an even cycle decomposition of index 3, i.e., the edge-set of L(G) can be partitioned into three 2-regular subgraphs whose connected components are cycles of even length. The more general problem of the existence of even cycle decompositions of index m in 2d-regular graphs is also addressed

Parallelism

EnProblems involving the idea of parallelism occur in finite geometry and in graph theory. This article addresses the question of constructing parallelisms with some degree of "symmetry". In particular, can we say anything on parallelisms admitting an automorphism group acting doubly transitively on "parallel classes"

Balance, partial balance and balanced-type spectra in graph-designs

For a given graph G, the set of positive integers v for which a G-design exists is usually called the 'spectrum' for G and the determination of the spectrum is sometimes called the 'spectrum problem'. We consider the spectrum problem for G-designs satisfying additional conditions of 'balance', in the case where G is a member of one of the following infinite families of trees: caterpillars, stars, comets, lobsters and trees of diameter at most 5. We determine the existence spectrum for balanced G-designs, degree-balanced and partially degree-balanced G-designs, orbit-balanced G-designs. We also address the existence question for non-balanced G-designs, for G-designs which are either balanced or partially degree-balanced but not degree-balanced, for G-designs which are degree-balanced but not orbit-balanced

Graph Decompositions and Symmetry

In this paper I shall try to review some results which were obtained in the area of factorizations and decompositions of complete graphs admitting an automorphism group with some specified properties. These properties primarily involve the action of the group on the objects of the decomposition, most oftenvertices, but also edges, subgraphs of the decomposition or factors of the factorization.Classification theorems were obtained in highly symmetric situations, for example when the group acts doubly transitively on vertices, and it is often the case that all examples arise from geometry in this context.A “less” symmetric situation involves a group acting sharply transitively on vertices, which means for any two given vertices there exists precisely one group element mapping the first vertex to the second one. The vertices of the complete graph can be identified with group elements in this case, and the decompositionor factorization can be described entirely within the group by techniques which are generally known as “difference” or “starter-like” methods. Existence may be a non-trivial question and generally depends on the isomorphism type of the chosen group