Gy\'arf\'as and Lehel and independently Faudree and Schelp proved that in any 2-coloring of the edges of $K_{n,n}$ there exists a monochromatic path on at least $2\lceil n/2\rceil$ vertices, and this is tight. We prove a stability version of this result which holds even if the host graph is not complete; that is, if $G$ is a balanced bipartite graph on $2n$ vertices with minimum degree at least $(3/4+o(1))n$, then in every 2-coloring of the edges of $G$, either there exists a monochromatic cycle on at least $(1+o(1))n$ vertices, or the coloring of $G$ is close to an extremal coloring -- in which case $G$ has a monochromatic path on at least $2\lceil n/2\rceil$ vertices and a monochromatic cycle on at least $2\lfloor n/2\rfloor$ vertices. Furthermore, we determine an asymptotically tight bound on the length of a longest monochromatic cycle in a 2-colored balanced bipartite graph on $2n$ vertices with minimum degree $\delta n$ for all $0\leq \delta\leq 1$.

中文翻译：

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2 色二分图中的长单色路径和循环

Gy\'arf\'as 和 Lehel 以及独立的 Faudree 和 Schelp 证明了在 $K_{n,n}$ 边缘的任何 2-着色中，至少在 $2\lceil n/2\rceil$ 上存在单色路径顶点，这是紧的。我们证明了这个结果的稳定性版本，即使主机图不完整也能成立；也就是说，如果 $G$ 是 $2n$ 顶点上的平衡二部图，最小度数至少为 $(3/4+o(1))n$，则在 $G$ 边的每 2 次着色中，要么在至少 $(1+o(1))n$ 个顶点上存在单色循环，要么 $G$ 的着色接近极值着色——在这种情况下，$G$ 至少在$2\lceil n/2\rceil$ 顶点和至少 $2\lfloor n/2\rfloor$ 顶点上的单色循环。此外，