Monochromatic cycles in 2-edge-colored bipartite graphs with large minimum degree

Abstract

For graphs G, $G_1$ and $G_2$, we write $G⟼(G_1,G_2)$ if any red-blue edge coloring of G yields a red $G_1$ or a blue $G_2$. The Ramsey number $r(G_1,G_2)$ is the minimum number n such that the complete graph $K_n⟼(G_1,G_2)$. There is an interesting phenomenon that for some graphs $G_1$ and $G_2$ there is a number $0<c<1$ such that for any graph G of order $r(G_1,G_2)$ with minimum degree $\delta \left(G\right)>c\left|V\right(G\left)\right|$, $G⟼(G_1,G_2)$. When we focus on bipartite graphs, the bipartite Ramsey number $br(G_1,G_2)$ is the minimum number n such that the complete bipartite graph $K_{n,n}⟼(G_1,G_2)$. Previous known related results on cycles are on the diagonal case ($G_1=G_2=C_{2n}$). In this paper, we obtain an asymptotically tight bound for all off-diagonal cases, namely, we determine an asymptotically tight bound on the minimum degree of a balanced bipartite graph G with order $br(C_{2m},C_{2n})$ in each part such that $G⟼(C_{2m},C_{2n})$. We show that for every $\eta >0$, there is an integer $N_0>0$ such that for any $N>N_0$ the following holds: Let ${\alpha }_1>{\alpha }_2>0$ such that ${\alpha }_1+{\alpha }_2=1$. Let $G[X,Y]$ be a balanced bipartite graph on $2(N-1)$ vertices with minimum degree $\delta \left(G\right)\ge (\frac{3}{4}+3\eta )(N-1)$. Then for any red-blue edge coloring of G, either there exist red even cycles of each length in $\{4,6,8,\dots ,(2-3{\eta }^2\left){\alpha }_{1}N\right\}$, or there exist blue even cycles of each length in $\{4,6,8,\dots ,(2-3{\eta }^2\left){\alpha }_{2}N\right\}$. A construction is given to show that the bound $\delta \left(G\right)\ge (\frac{3}{4}+3\eta )(N-1)$ is asymptotically tight. Furthermore, we give a stability result.