desarguesian projective plane
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2015 ◽  
Vol 107 (1) ◽  
pp. 89-117 ◽  
Author(s):  
Daniele Bartoli ◽  
Alexander A. Davydov ◽  
Giorgio Faina ◽  
Alexey A. Kreshchuk ◽  
Stefano Marcugini ◽  
...  


2014 ◽  
Vol 47 (4) ◽  
Author(s):  
Jan Jakóbowski ◽  
Danuta Kacperek

AbstractA. Lewandowski and H. Makowiecka proved in 1979 that existence of the Havlicek-Tietze configuration (shortly H-T) in the desarguesian projective plane is equivalent to existence in the associated field, a root of polynomial x



2014 ◽  
Vol 106 (1) ◽  
pp. 1-17 ◽  
Author(s):  
Daniele Bartoli ◽  
Alexander A. Davydov ◽  
Giorgio Faina ◽  
Stefano Marcugini ◽  
Fernanda Pambianco


2014 ◽  
Vol 144 ◽  
pp. 110-122 ◽  
Author(s):  
John Bamberg ◽  
Anton Betten ◽  
Cheryl E. Praeger ◽  
Alfred Wassermann


2013 ◽  
Vol 104 (1) ◽  
pp. 11-43 ◽  
Author(s):  
Daniele Bartoli ◽  
Alexander A. Davydov ◽  
Giorgio Faina ◽  
Stefano Marcugini ◽  
Fernanda Pambianco


10.37236/2582 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Tamás Héger ◽  
Marcella Takáts

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.



10.37236/107 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Hendrik Van Maldeghem

In this paper we improve on a result of Beutelspacher, De Vito & Lo Re, who characterized in 1995 finite semiaffine spaces by means of transversals and a condition on weak parallelism. Basically, we show that one can delete that condition completely. Moreover, we extend the result to the infinite case, showing that every plane of a planar space with at least two planes and such that all planes are semiaffine, comes from a (Desarguesian) projective plane by deleting either a line and all of its points, a line and all but one of its points, a point, or nothing.



10.37236/1120 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Nick Gill

Let $N$ be a nilpotent group normal in a group $G$. Suppose that $G$ acts transitively upon the points of a finite non-Desarguesian projective plane ${\cal P}$. We prove that, if ${\cal P}$ has square order, then $N$ must act semi-regularly on ${\cal P}$. In addition we prove that if a finite non-Desarguesian projective plane ${\cal P}$ admits more than one nilpotent group which is regular on the points of ${\cal P}$ then ${\cal P}$ has non-square order and the automorphism group of ${\cal P}$ has odd order.





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