On a conjecture of Erdos and Simonovits: Even Cycles

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.