An Extremal Characterization of Projective Planes
Jacques VerstraeteThe Electronic Journal of Combinatorics
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Abstract
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.
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