On the number of $k$-gons in finite projective planes

Abstract

Let $\Pi$ be a projective plane of order $n$ and ${\Gamma }_{\Pi }$ be its Levi graph (the point-line incidence graph). For fixed $k\ge 3$, let $c_{2k}({\Gamma }_{\Pi })$ denote the number of $2k$-cycles in ${\Gamma }_{\Pi }$. In this paper we show that $$c_{2k}({\Gamma }_{\Pi })=\frac{1}{2k}{n}^{2k}+O(n^{2k-2}),n\to \infty .$$ We also state a conjecture regarding the third and fourth largest terms in the asymptotic of the number of $2k$-cycles in ${\Gamma }_{\Pi }$. This result was also obtained independently by Voropaev in 2012. Let $\text{ex}(v,C_{2k},{\mathcal{C}}_{\text{odd}}\cup \{C_4\})$ denote the greatest number of $2k$-cycles amongst all bipartite graphs of order $v$ and girth at least 6. As a corollary of the result above, we obtain $$\text{ex}(v,C_{2k},{\mathcal{C}}_{\text{odd}}\cup \{C_4\})=\left(\frac{1}{2^{k+1}k}-o(1)\right)v^k,v\to \infty .$$