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

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