The 2-Blocking Number and the Upper Chromatic Number of PG(2,q)

Gábor Bacsó, Tamás Héger, T. Szőnyi

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

A twofold blocking set (double blocking set) in a finite projective plane Π is a set of points, intersecting every line in at least two points. The minimum number of points in a double blocking set of Π is denoted by τ2(Π). Let PG(2,q) be the Desarguesian projective plane over GF(q), the finite field of q elements. We show that if q is odd, not a prime, and r is the order of the largest proper subfield of GF(q), then τ2PG(2,q))≤ 2(q+(q-1)/(r-1)). For a finite projective plane Π, let χ-(Π) denote the maximum number of classes in a partition of the point-set, such that each line has at least two points in some partition class. It can easily be seen that χ-(Π)≥v-τ2(Π)+1 ({star operator}) for every plane Π on v points. Let q=ph, p prime. We prove that for Π= PG (2,q), equality holds in ({star operator}) if q and p are large enough.

Original languageEnglish
JournalJournal of Combinatorial Designs
DOIs
Publication statusAccepted/In press - 2013

Fingerprint

Chromatic number
Blocking Set
Finite projective plane
Star
Partition
Line
Subfield
Projective plane
Operator
Point Sets
Set of points
Galois field
Equality
Odd
Denote
Class

Keywords

  • 05B25
  • Double blocking set
  • Finite projective plane
  • Hypergraph
  • MSC2000 Subject Classification: 05C15
  • Upper chromatic number

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics

Cite this

The 2-Blocking Number and the Upper Chromatic Number of PG(2,q). / Bacsó, Gábor; Héger, Tamás; Szőnyi, T.

In: Journal of Combinatorial Designs, 2013.

Research output: Contribution to journalArticle

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N2 - A twofold blocking set (double blocking set) in a finite projective plane Π is a set of points, intersecting every line in at least two points. The minimum number of points in a double blocking set of Π is denoted by τ2(Π). Let PG(2,q) be the Desarguesian projective plane over GF(q), the finite field of q elements. We show that if q is odd, not a prime, and r is the order of the largest proper subfield of GF(q), then τ2PG(2,q))≤ 2(q+(q-1)/(r-1)). For a finite projective plane Π, let χ-(Π) denote the maximum number of classes in a partition of the point-set, such that each line has at least two points in some partition class. It can easily be seen that χ-(Π)≥v-τ2(Π)+1 ({star operator}) for every plane Π on v points. Let q=ph, p prime. We prove that for Π= PG (2,q), equality holds in ({star operator}) if q and p are large enough.

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