On a conjecture of Erdos and Simonovits: Even Cycles

TL;DR

This paper investigates extremal graph theory related to bipartite graphs avoiding certain cycles, extending previous results and identifying conditions under which extremal graphs are bipartite incidence graphs of generalized polygons.

Contribution

It extends Erdős and Simonovits's conjecture by establishing asymptotic equivalences for Turán numbers involving cycle families and characterizes extremal graphs for specific cycle lengths.

Findings

For certain cycle lengths, extremal graphs are asymptotically bipartite incidence graphs of generalized polygons.

The asymptotic equivalence holds when the cycle length exceeds twice the smaller cycle length.

Exact extremal structures are identified for infinitely many cases, with limitations for smaller cycle lengths.

Abstract

Let $\mc{F}$ be a family of graphs. A graph is {\em $\mc{F}$-free} if it contains no copy of a graph in $\mc{F}$ as a subgraph. A cornerstone of extremal graph theory is the study of the {\em Tur\'an number} $ex(n,\mc{F})$, the maximum number of edges in an $\mc{F}$-free graph on $n$ vertices. Define the {\em Zarankiewicz number} $z(n,\mc{F})$ to be the maximum number of edges in an $\mc{F}$-free {\em bipartite} graph on $n$ vertices. Let $C_k$ denote a cycle of length $k$, and let $\mc{C}_k$ denote the set of cycles $C_{\ell}$, where $3 \le \ell \leq k$ and $\ell$ and $k$ have the same parity. Erd\H{o}s and Simonovits conjectured that for any family $\mc{F}$ consisting of bipartite graphs there exists an odd integer $k$ such that $ex(n,\mc{F} \cup \mc{C}_k) \sim z(n,\mc{F})$. They proved this when $\mc{F}={C_4}$ by showing that $ex(n,\{C_4,C_5\}) \sim z(n,C_4)$. In this paper, we extend this result by showing that if $\ell \in \{2,3,5\}$ and $k > 2\ell$ is odd, then ${ex(n,\mc{C}_{2\ell} \cup {C_k}) \sim z(n,\mc{C}_{2\ell})$. Furthermore, if $k > 2\ell + 2$ is odd, then for infinitely many $n$ we show that the extremal $\mc{C}_{2\ell} \cup \{C_k\}$-free graphs are bipartite incidence graphs of generalized polygons. We observe that this exact result does not hold for any odd $k < 2\ell$, and furthermore the asymptotic result does not hold when $(\ell,k)$ is $(3,3)$, $(5,3)$ or $(5,5)$. Our proofs make use of pseudorandomness properties of nearly extremal graphs that are of independent interest.