An Extremal Characterization of the Incidence Graphs of Projective Planes

Abstract

Let G be a 4-cycle free, bipartite graph on 2n vertices with partitions of equal cardinality n. Let c6(G) denote the number of cycles of length 6 in G. We prove that for n ≥ 3, c6(G) ≤ \(\frac{1} {3}\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right)(n - r_n ) \), where \(r_n = \frac{1} {2} + \frac{{\sqrt {4n - 3} }} {2} \) , with equality if and only if G is the incidence point-line graph of a projective plane.