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

TL;DR

This paper analyzes the asymptotic number of 2k-cycles in the Levi graph of finite projective planes, providing precise estimates and conjectures, and relates these findings to extremal cycle counts in bipartite graphs.

Contribution

It establishes the asymptotic count of 2k-cycles in Levi graphs of projective planes and connects this to extremal cycle problems in bipartite graphs, including a conjecture on higher-order terms.

Findings

Number of 2k-cycles in Levi graphs is approximately (1/2k)n^{2k} for large n.

Derived bounds for the maximum number of 2k-cycles in bipartite graphs with girth at least 6.

Connected cycle counts in projective planes to extremal graph theory problems.

Abstract

Let $\Pi$ be a projective plane of order $n$ and $\Gamma_{\Pi}$ be its Levi graph (the point-line incidence graph). For fixed $k \geq 3$, let $c_{2k}(\Gamma_{\Pi})$ denote the number of $2k$-cycles in $\Gamma_{\Pi}$. In this paper we show that $$ c_{2k}(\Gamma_{\Pi}) = \frac{1}{2k}n^{2k} + O(n^{2k-2}), \hspace{0.5cm} n \rightarrow \infty. $$ 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 $$ \text{ex}(v, C_{2k}, \mathcal{C}_{\text{odd}}\cup \{C_4\}) = \left(\frac{1}{2^{k+1}k}-o(1)\right)v^k, \hspace{0.5cm} v \rightarrow \infty. $$