On a conjecture of Erdős and Simonovits: Even cycles

• Authors: Peter Keevash;Benny Sudakov;Jacques Verstraëte
• Venue: Combinatorica
• Year: 2013

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. Erd驴s and Simonovits conjectured that for any family F consisting of bipartite graphs there exists an odd integer k such that ex(n,F 驴 C k ) ~ z(n,F) -- here we write f(n) ~ g(n) for functions f,g: 驴 驴 驴 if lim n驴驴 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驴{2,3,5} and k2ℓ is odd, then ex(n,C 2ℓ 驴{C k }) ~ z(n,C 2ℓ ). Furthermore, if k2ℓ+2 is odd, then for infinitely many n we show that the extremal C 2ℓ 驴{C k }-free graphs are bipartite incidence graphs of generalized polygons. We observe that this exact result does not hold for any odd kℓ,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.