György Kiss

On the spectrum of the sizes of semiovals in PG(2,q) , q odd

Some characterization theorems and non-existence results of semiovals with extra properties are proved. New examples of large semiovals are constructed for q=11 and q=13.

On the X-ray number of almost smooth convex bodies and of convex bodies of constant width

The X-ray numbers of some classes of convex bodies are investigated. In particular, we give a proof of the X-ray Conjecture as well as of the Illumination Conjecture for almost smooth convex bodies of any dimension and for convex bodies of constant width of dimensions 3, 4, 5 and 6.

The Cyclic Model forPG(n, q) and a Construction of Arcs

The n -dimensional finite projective space, PG(n, q), admits a cyclic model, in which the set of points of PG(n, q) is identified with the elements of the group Zqn+qn?1+?+q+1. It was proved by Hall (1974, Math. Centre Tracts, 57, 1?26) that in the cyclic model of PG(2, q), the additive inverse of a line is a conic. The following generalization of this result is proved:In the cyclic model of PG(n, q), the additive inverse of a line is a (q+ 1)-arc if n+ 1 is a prime and q+ 1 n.It is also shown that the additive inverse of a line is always a normal rational curve in some subspace PG(m, q), where m+ 1| n+ 1.

On the successive illumination parameters of convex bodies

SummaryThe notion of successive illumination parameters of convex bodies is introduced. We prove some theorems in the plane and determine the exact values of the successive illumination parameters of spheres, cubes and cross-polytopes for some dimensions.

A note on m-factorizations of complete multigraphs arising from designs

Some new infinite families of simple, indecomposable m -factorizations of the complete multigraph λK v are presented. Most of the constructions come from finite geometries.

Shape-regular polygons in finite planes

The notion of shape in the Gaussian plane was introduced by Lester [5] and extended by Artzy [1]. In this paper we generalize this notion in the affine planesAG(2,q) over the Galois fieldGF(q), q=pr andp an odd prime. We investigate the existence of shape-regular polygons and the correspondence between shape-regularity and affine-regularity.

Tangent sets in finite spaces

Abstract We define the tangent set of a given point set of PG( n , q ) and give some examples. The tangent sets of hyperplanes, arcs, quadrics and Hermitian varieties are investigated.

Large antipodal families

A family {Ai | i ∈ I} of sets in ℝd is antipodal if for any distinct i, j ∈ I and any p ∈ Ai, q ∈ Aj, there is a linear functional ϕ:ℝd → ℝ such that ϕ(p) ≠ ϕ(q) and ϕ(p) ≤ ϕ(r) ≤ ϕ(q) for all r ∈ ∪i∈IAi. We study the existence of antipodal families of large finite or infinite sets in ℝ3.

Two generalization of Napoleon's theorem in finite planes

The following theorem about triangles in the Euclidean plane is attributed to Napoleon: Let A=A1A2A3 be a triangle in the Euclidean plane and B=B1B2B3 be the triangle whose vertices are the centers of the equilateral triangles all erected externally (or all internally) on the sides of A. Then B is an equilateral triangle. Two generalizations of this theorem in Galois planes of odd order are given. The proofs are based on an algebraic method which was developed by Bachmann and Schmidt (n-Ecke, Hochschultaschenbucher Verlag, Mannheim, Wein, Zurich, 1970) and Fisher et al. (The Geometric Vein, Springer, New York, 1981, pp. 321–333) to deal with geometry problems.

Edge-girth-regular graphs

Abstract We consider a new type of regularity we call edge-girth-regularity. An edge-girth-regular ( v , k , g , λ ) -graph Γ is a k -regular graph of order v and girth g in which every edge is contained in λ distinct g -cycles. This concept is a generalization of the well-known concept of ( v , k , λ ) -edge-regular graphs (that count the number of triangles) and appears in several related problems such as Moore graphs and Cage and Degree/Diameter Problems. All edge- and arc-transitive graphs are edge-girth-regular as well. We derive a number of basic properties of edge-girth-regular graphs, systematically consider cubic and tetravalent graphs from this class, and introduce several constructions that produce infinite families of edge-girth-regular graphs. We also exhibit several surprising connections to regular embeddings of graphs in orientable surfaces.