Geodesic
An extremal characterization of projective planes
The electronic journal of combinatorics, Tome 15 (2008)

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Zbl EuDML
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.
DOI : 10.37236/867
Classification : 05B25, 05C35, 05C38, 51E14, 90C30
Mots-clés : bipartite graphs, incidence graph, projective plane, maximum number of cycles, number of quadrilaterals 
Stefaan De Winter; Felix Lazebnik; Jacques Verstraëte. An extremal characterization of projective planes. The electronic journal of combinatorics, Tome 15 (2008). doi: 10.37236/867
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     author = {Stefaan De Winter and Felix Lazebnik and Jacques Verstra\"ete},
     title = {An extremal characterization of projective planes},
     journal = {The electronic journal of combinatorics},
     year = {2008},
     volume = {15},
     doi = {10.37236/867},
     zbl = {1178.05024},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/867/}
}
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