DeBiasio and Krueger showed the following result: For all 0 ≤ δ ≤ 1 and ϵ > 0, there exists n0 such that if G is a balanced bipartite graph on 2n ≥ 2n0 vertices with δ(G) = δn, then in every 2-coloring of G, there exists a monochromatic cycle of order at least (f(δ) − ϵ)n, where \(f(\delta)=\begin{cases}{\delta}, & {0 \leq \delta \leq {2 \over 3}},\\{4{\delta}-2}, & {{2 \over 3} < \delta \leq {3 \over 4}},\\1, & {3 \over 4} < \delta \leq 1.\end{cases}\) Zhang and Peng (2023) extended the above result to off-diagonal cases when \({\delta} > {3 \over 4}\) . In this paper, we relax the condition \({\delta} > {3 \over 4}\) to \({\delta} > {2 \over 3}\) . We show the following result: For every η > 0, there exists a positive integer N0 such that for every integer N > N0 the following holds. Let \({2 \over 3} < {\delta} \leq {3 \over 4}\) , and let \({\alpha_1} \geq {{\delta\alpha}_{2} \over {3\delta - 2}} > 0\) such that α1 + α2 = 1. Let G[X, Y] be a balanced bipartite graph on 2N vertices with δ(G) = (δ + 3η)N. Then for each red-blue-edge-coloring of G, either there exist red even cycles of each length in {4, 6, 8, …, 2(2δ − 1)(2 − 3η2)α1N}, or there exist blue even cycles of each length in {4, 6, 8, …, 2(2δ − 1)(2 − 3δ2)α2N}. There are constructions of colorings showing that the length of a longest monochromatic cycle is asymptotically tight and the condition \({\alpha_1} \geq {{\delta\alpha}_{2} \over {3\delta - 2}}\) cannot be removed.
