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Constructions of bipartite graphs from finite geometries
, 2005
"... We construct an incidence structure using certain points and lines in finite projective spaces. The structural properties of the associated bipartite incidence graphs are analyzed. These n × n bipartite graphs provide constructions of C6free graphs with Ω(n 4/3) edges, C10free graphs with Ω(n 6/5) ..."
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We construct an incidence structure using certain points and lines in finite projective spaces. The structural properties of the associated bipartite incidence graphs are analyzed. These n × n bipartite graphs provide constructions of C6free graphs with Ω(n 4/3) edges, C10free graphs with Ω(n 6/5) edges, and Θ(7, 7, 7)free graphs with Ω(n 8/7) edges. Each of these bounds is sharp in order of magnitude.
Extensions of Extremal Graph Theory to Grids
"... We consider extensions of Turán’s original theorem of 1941 to planar grids. For a complete k x m array of vertices, we establish in Proposition 4.3 an exact formula for the maximal number of edges possible without any square regions. We establish with Theorem 4.12 an upper bound and with Theorem 4.1 ..."
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We consider extensions of Turán’s original theorem of 1941 to planar grids. For a complete k x m array of vertices, we establish in Proposition 4.3 an exact formula for the maximal number of edges possible without any square regions. We establish with Theorem 4.12 an upper bound and with Theorem 4.15 an asymptotic lower bound for the maximal number of edges on a general grid graph with n vertices and no rectangles.
Contributions to Pure and Applicable Galois Geometry
, 2012
"... tot het behalen van de graad van Doctor in de Wetenschappen: Wiskunde. Preface The term Galois geometry originates from an article by Segre [86], wherein he refers to a finite projective plane as Galois plane. Later, Hirschfeld and Thas in their book ”General Galois geometries” [44] denominate finit ..."
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tot het behalen van de graad van Doctor in de Wetenschappen: Wiskunde. Preface The term Galois geometry originates from an article by Segre [86], wherein he refers to a finite projective plane as Galois plane. Later, Hirschfeld and Thas in their book ”General Galois geometries” [44] denominate finite projective spaces as Galois geometries. Indeed, both Segre, and Hirschfeld and Thas are united in their desire of emphasizing that an analytical approach to finite projective geometry is predicated on finite or Galois fields and their (Galois) extensions, thus recognising the important contributions made by the famous French mathematician É. Galois (18111832) in algebra. All geometries discussed in this thesis are finite and can be constructed in a finite projective space, such as generalised quadrangles (Chapter 1, Section 1.2) or as egglike inversive planes (Chapter 1, Section 1.4). In the case of inversive planes, the more common algebraic constructions are based on finite fields and their cubic extensions. Indeed, one can view Galois geometry as the concept that encompasses all analytical geometries over a finite field and its extensions [33]. In Chapter 1 the relevant definitions and theorems for these geometries are gathered for reference later on.