scholarly journals An Extremal Characterization of Projective Planes

10.37236/867 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Stefaan De Winter ◽  
Felix Lazebnik ◽  
Jacques Verstraëte
Keyword(s):  
Projective Plane ◽  
Projective Planes ◽  
Bipartite Graphs ◽  
Incidence Graph ◽  
Number Of Cycles

In this article, we prove that amongst all $n$ by $n$ bipartite graphs of girth at least six, where $n = q^2 + q + 1 \ge 157$, the incidence graph of a projective plane of order $q$, when it exists, has the maximum number of cycles of length eight. This characterizes projective planes as the partial planes with the maximum number of quadrilaterals.

scholarly journals Resolving Sets and Semi-Resolving Sets in Finite Projective Planes

10.37236/2582 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Tamás Héger ◽  
Marcella Takáts
Keyword(s):  
Projective Plane ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
The Other ◽  
Blocking Set ◽  
Metric Dimension ◽  
Desarguesian Projective Plane ◽  
Resolving Set ◽  
Incidence Graph ◽  
Vertex Set

In a graph $\Gamma=(V,E)$ a vertex $v$ is resolved by a vertex-set $S=\{v_1,\ldots,v_n\}$ if its (ordered) distance list with respect to $S$, $(d(v,v_1),\ldots,d(v,v_n))$, is unique. A set $A\subset V$ is resolved by $S$ if all its elements are resolved by $S$. $S$ is a resolving set in $\Gamma$ if it resolves $V$. The metric dimension of $\Gamma$ is the size of the smallest resolving set in it. In a bipartite graph a semi-resolving set is a set of vertices in one of the vertex classes that resolves the other class.We show that the metric dimension of the incidence graph of a finite projective plane of order $q\geq 23$ is $4q-4$, and describe all resolving sets of that size. Let $\tau_2$ denote the size of the smallest double blocking set in PG$(2,q)$, the Desarguesian projective plane of order $q$. We prove that for a semi-resolving set $S$ in the incidence graph of PG$(2,q)$, $|S|\geq \min \{2q+q/4-3, \tau_2-2\}$ holds. In particular, if $q\geq9$ is a square, then the smallest semi-resolving set in PG$(2,q)$ has size $2q+2\sqrt{q}$. As a corollary, we get that a blocking semioval in PG$(2, q)$, $q\geq 4$, has at least $9q/4-3$ points. A corrigendum was added to this paper on March 3, 2017.

scholarly journals The Endomorphism Type of Certain Bipartite Graphs and a Characterization of Projective Planes

2011 ◽  
Vol 29 (2) ◽  
pp. 253-257
Author(s):  
Ralf Köhl ◽  
Katja Möser ◽  
Hendrik Van Maldeghem
Keyword(s):  
Projective Planes ◽  
Bipartite Graphs

scholarly journals Homeomorphisms of Quaternion space and projective planes in four space

1977 ◽  
Vol 23 (1) ◽  
pp. 112-128 ◽  
Author(s):  
T. M. Price
Keyword(s):  
Projective Plane ◽  
Fundamental Group ◽  
Projective Planes ◽  
Finite Fundamental Group ◽  
Quaternion Space

AbstractIt is known that all locally flat projective planes in S4 have homeomorphic normal disk bundles. In this paper we investigate the homeomorphisms of Q3 (= boundary of the normal disk bundle) on to itself. We show that a homeomorphisms of Q3 onto itself is determined, up to isotopy, by the outerautomorphism of π1(Q3) that it induces. Since Q3 is an irreducible, not sufficiently large 3-manifold with finite fundamental group this characterization is interesting in its own right. The characterization of homeomorphisms is then used to study certain questions about embeddings of the projective plane in S4. One result is that there are at most two distinct projective planes in S4 with a given complement.

scholarly journals Maximal arcs in projective planes of order 16 and related designs

2019 ◽  
Vol 19 (3) ◽  
pp. 345-351 ◽  
Author(s):  
Mustafa Gezek ◽  
Vladimir D. Tonchev ◽  
Tim Wagner
Keyword(s):  
Projective Plane ◽  
Projective Planes ◽  
Maximal Arcs ◽  
Even Order

Abstract The resolutions and maximal sets of compatible resolutions of all 2-(120,8,1) designs arising from maximal (120,8)-arcs, and the 2-(52,4,1) designs arising from previously known maximal (52,4)-arcs, as well as some newly discovered maximal (52,4)-arcs in the known projective planes of order 16, are computed. It is shown that each 2-(120,8,1) design associated with a maximal (120,8)-arc is embeddable in a unique way in a projective plane of order 16. This result suggests a possible strengthening of the Bose–Shrikhande theorem about the embeddability of the complement of a hyperoval in a projective plane of even order. The computations of the maximal sets of compatible resolutions of the 2-(52,4,1) designs associated with maximal (52,4)-arcs show that five of the known projective planes of order 16 contain maximal arcs whose associated designs are embeddable in two nonisomorphic planes of order 16.

A Characterization of Some Geometries of Chains

1974 ◽  
Vol 26 (02) ◽  
pp. 257-272 ◽  
Author(s):  
Yi Chen
Keyword(s):  
Projective Plane ◽  
Minkowski Plane ◽  
Close Relation ◽  
Laguerre Plane ◽  
Geometric Structures ◽  
Möbius Plane

The geometries considered here are the Möbius plane M() (W. Benz [1]), the Laguerre plane L() (W. Benz and H. Mäurer [7]) and the Minkowski plane A() (W. Benz [5], G. Kaerlein [18]) over a field . All of them are geometries of an algebra with identity over a field. The characterization of the projective plane over a field by the proposition of Pappus first gave a close relation between algebraic and geometric structures. B. L. v. d. Waedern and L. J. Smid [28] presented a further example by characterizing the Möbius and Laguerre plane with incidence axioms and the "complete" proposition of Miquel.

scholarly journals Analysis of Some Properties from Construction of Finite Projective Planes of Order 2 and 3

CAUCHY ◽  
2016 ◽  
Vol 4 (3) ◽  
pp. 131
Author(s):  
Vira Hari Krisnawati ◽  
Corina Karim
Keyword(s):  
Automorphism Group ◽  
Projective Plane ◽  
Symmetric Group ◽  
Block Design ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Steiner System ◽  
Incidence Relation ◽  
Finite Projective Planes ◽  
Set Of Points

<p class="abstract"><span lang="IN">In combinatorial mathematics, a Steiner system is a type of block design. Specifically, a Steiner system <em>S</em>(<em>t</em>, <em>k</em>, <em>v</em>) is a set of <em>v</em> points and <em>k</em> blocks which satisfy that every <em>t</em>-subset of <em>v</em>-set of points appear in the unique block. It is well-known that a finite projective plane is one examples of Steiner system with <em>t</em> = 2, which consists of a set of points and lines together with an incidence relation between them and order 2 is the smallest order.</span></p><p class="abstract"><span lang="IN">In this paper, we observe some properties from construction of finite projective planes of order 2 and 3. Also, we analyse the intersection between two projective planes by using some characteristics of the construction and orbit of projective planes over some representative cosets from automorphism group in the appropriate symmetric group.</span></p>

Projective planes of infinite but isolic order

1976 ◽  
Vol 41 (2) ◽  
pp. 391-404 ◽  
Author(s):  
J. C. E. Dekker
Keyword(s):  
Projective Plane ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Latin Squares ◽  
Combinatorial Theory ◽  
Ternary Ring ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
Planar Ternary Ring ◽  
Partial Recursive Functions

The main purpose of this paper is to show how partial recursive functions and isols can be used to generalize the following three well-known theorems of combinatorial theory.(I) For every finite projective plane Π there is a unique number n such that Π has exactly n2 + n + 1 points and exactly n2 + n + 1 lines.(II) Every finite projective plane of order n can be coordinatized by a finite planar ternary ring of order n. Conversely, every finite planar ternary ring of order n coordinatizes a finite projective plane of order n.(III) There exists a finite projective plane of order n if and only if there exist n − 1 mutually orthogonal Latin squares of order n.

Weakly Isotopic Planar Ternary Rings

1975 ◽  
Vol 27 (1) ◽  
pp. 32-36
Author(s):  
Frederick W. Stevenson
Keyword(s):  
Projective Plane ◽  
Projective Planes ◽  
Ternary Ring ◽  
Algebraic Relation ◽  
Planar Ternary Ring ◽  
Algebraic Properties

This paper introduces two relations both weaker than isotopism which may hold between planar ternary rings. We will concentrate on the geometric consequences rather than the algebraic properties of these relations. It is well-known that every projective plane can be coordinatized by a planar ternary ring and every planar ternary ring coordinatizes a projective plane. If two planar ternary rings are isomorphic then their associated projective planes are isomorphic; however, the converse is not true. In fact, an algebraic bond which necessarily holds between the coordinatizing planar ternary rings of isomorphic projective planes has not been found. Such a bond must, of course, be weaker than isomorphism; furthermore, it must be weaker than isotopism. Here we show that it is even weaker than the two new relations introduced.This is significant because the weaker of our relations is, in a sense, the weakest possible algebraic relation which can hold between planar ternary rings which coordinatize isomorphic projective planes.

A Characterization of the Hughes Planes

1965 ◽  
Vol 17 ◽  
pp. 916-922 ◽  
Author(s):  
T. G. Ostrom
Keyword(s):  
Fixed Point ◽  
Projective Plane ◽  
Related Notion ◽  
Fixed Line

Baer (1) introduced the term "(p,L)-collineation" to denote a central collineation with centre p and axis L. We shall find it convenient to use a modification of the related notion of "(p, L)-transitivity."Definition. Let π0 be a subplane of the projective plane π. Let L be a fixed line of π0, and let p be a fixed point of π0. Let r and s be any two points of π0 that are collinear with p, distinct from p, and not on L. If, for each such choice of r and s, there is a (p, L)-collineation of π that (1) carries π0 into itself and (2) carries r into s, we shall say that π is (p, L, π0)-transitive.

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