Let the bipartite Tur\'an number $ex(m,n,H)$ of a graph $H$ be the maximum number of edges in an $H$-free bipartite graph with two parts of sizes $m$ and $n$, respectively. In this paper, we prove that $ex(m,n,C_{ 2t})=(t-1)n+m-t+1$ for any positive integers $m,n,t$ with $n\geq m\geq t\geq \frac{ m}{ 2}+1$. This confirms the rest of a conjecture of Gy\"{ o}ri \cite{ G97} (in a stronger form), and improves the upper bound of $ex(m,n,C_{ 2t})$ obtained by Jiang and Ma \cite{ JM18} for this range. We also prove a tight edge condition for consecutive even cycles in bipartite graphs, which settles a conjecture in \cite{ A09}. As a main tool, for a longest cycle $C$ in a bipartite graph, we obtain an estimate on the upper bound of the number of edges which are incident to at most one vertex in $C$. Our two results generalize or sharpen a classical theorem due to Jackson \cite{ J85} in different ways.
Categories: math.CO