On a conjecture of Erdős and Simonovits: Even cycles

Abstract

Let F be a family of graphs. A graph is F-free if it contains no copy of a graph in F as a subgraph. A cornerstone of extremal graph theory is the study of the Turán number ex(n,F), the maximum number of edges in an F-free graph on n vertices. Define the Zarankiewicz number z(n,F) to be the maximum number of edges in an F-free bipartite graph on n vertices. Let C k denote a cycle of length k, and let C k denote the set of cycles C ℓ, where 3≤ℓ≤k and ℓ and k have the same parity. Erdős and Simonovits conjectured that for any family F consisting of bipartite graphs there exists an odd integer k such that ex(n,F ∪ C k ) ∼ z(n,F) — here we write f(n) ∼ g(n) for functions f,g: ℕ → ℝ if lim n→∞ f(n)/g(n)=1. They proved this when F ={C 4} by showing that ex(n,{C 4;C 5})∼z(n,C 4). In this paper, we extend this result by showing that if ℓ∈{2,3,5} and k>2ℓ is odd, then ex(n,C 2ℓ ∪{C k }) ∼ z(n,C 2ℓ ). Furthermore, if k>2ℓ+2 is odd, then for infinitely many n we show that the extremal C 2ℓ ∪{C k }-free graphs are bipartite incidence graphs of generalized polygons. We observe that this exact result does not hold for any odd k<2ℓ, and furthermore the asymptotic result does not hold when (ℓ,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.