The bipartite Turán number, denoted by $\text{ex}\left(m,n,H\right)$, is the maximum number of edges in an $H$-free bipartite graph with two parts of sizes $m$ and $n$, respectively. In this article, we prove $\text{ex}\left(m,n,C_{2t}\right)=\left(t-1\right)n+m-t+1$ for any positive integers $m,n,t$ with $n\ge m\ge t\ge \frac{m}{2}+1$, which confirms (in a strong form) the unsolved case of a conjecture of Győri. Let $G=\left(X,Y;E\right)$ be a bipartite graph with $\mid X\mid =n$ and $\mid Y\mid =m$. Suppose that $n\ge m\ge 2k+2$ for some $k\in \mathbb{N}$. We prove that $G$ contains cycles of all even lengths from 4 to $2m-2k$ if $e\left(G\right)\ge \left(m-k-1\right)n+k+2$. This result sharpens a theorem on long cycles in bipartite graphs due to Jackson. As a main tool, for a longest cycle $C$ in a bipartite graph, we prove an upper bound of the number of edges which are incident with at most one vertex in $C$, which is a bipartite analogue of a classical theorem of Bondy. Our proof technique is structural.

中文翻译：

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大偶数循环的精确二分图兰数

二分图兰数，表示为 $\text{前任}\left(米,n,H\right)$, 是最大边数 $H$- 两部分大小的自由二部图 $米$ 和 $n$， 分别。在本文中，我们证明$\text{前任}\left(米,n,C_{2吨}\right)=\left(吨-1\right)n+米-吨+1$ 对于任何正整数 $米,n,吨$ 和 $n\ge 米\ge 吨\ge \frac{米}{2}+1$，这证实了（以强形式）Győri 猜想的未解决情况。让$G=\left(X,是;乙\right)$ 是一个二部图 $\mid X\mid =n$ 和 $\mid 是\mid =米$. 假设$n\ge 米\ge 2克+2$ 对于一些 $克\in \mathbb{N}$. 我们证明$G$ 包含所有偶数长度的循环，从 4 到 $2米-2克$ 如果 $\mathrm{电子}\left(G\right)\ge \left(米-克-1\right)n+克+2$. 由于杰克逊，这个结果使二部图中的长周期定理更加尖锐。作为主要工具，周期最长$C$ 在二部图中，我们证明了最多与一个顶点相交的边数的上限 $C$，这是邦迪经典定理的二部类比。我们的证明技术是结构性的。