Long monochromatic paths and cycles in 2-colored bipartite graphs

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 $2\lceil n\mathrm{/}2\rceil$ 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 $(3\mathrm{/}4+o(1))n$, 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 $2\lceil n\mathrm{/}2\rceil$ vertices and a monochromatic cycle on at least $2\lfloor n\mathrm{/}2\rfloor$ 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 $\delta n$forall$0\leq \delta\leq 1$.