Partitioning the flags of PG(2,q) into strong representative systems

Csaba Mengyán

Abstract


In this paper we show

a natural extension of the idea used by Ill\'es, Sz\H{o}nyi and Wettl

\cite{swi} which proved that the flags of

${\rm PG} (2,q)$ can be partitioned into

$(q-1)\sqrt q+3q$ strong representative systems for $q$ an odd square.

From a generalization

of the Buekenhout construction of unitals \cite{kozoscikk} their idea

can be applied for any non-prime $q^h$ to yield

that $q^{2h-1}+2q^h$ strong representative systems partition

the flags of ${\rm PG} (2,q^{h})$.

In this way we

also give a solution to a question of Gy\'arf\'as \cite{FSGT} about

the strong chromatic index of the bipartite graph corresponding to

${\rm PG} (2,q)$, for $q$ non-prime.

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Contributions to Discrete Mathematics. ISSN: 1715-0868