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{\bf Stefaan De Winter, Felix Lazebnik and Jacques Verstra\"ete}
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{\bf An Extremal Characterization of Projective Planes}
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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.
\bye