Bipartite Ramsey numbers of large cycles

Abstract

For an integer $r\ge 2$ and bipartite graphs $H_i$, where $1\le i\le r$, the bipartite Ramsey number $br(H_1,H_2,\dots ,H_r)$ is the minimum integer $N$ such that any $r$-edge coloring of the complete bipartite graph $K_{N,N}$ contains a monochromatic subgraph isomorphic to $H_i$ in color $i$ for some $i$, $1\le i\le r$. We show that for ${\alpha }_1,{\alpha }_2>0$, $br(C_{2\lfloor {\alpha }_{1}n\rfloor },C_{2\lfloor {\alpha }_{2}n\rfloor })=({\alpha }_1+{\alpha }_2+o(1))n$. We also show that if $r\ge 3,{\alpha }_1,{\alpha }_2>0,{\alpha }_{j+2}\ge [(j+2)!-1]{\sum }_{i=1}^{j+1}{\alpha }_i$ for $j=1,2,\dots ,r-2$, then $br(C_{2\lfloor {\alpha }_{1}n\rfloor },C_{2\lfloor {\alpha }_{2}n\rfloor },\dots ,C_{2\lfloor {\alpha }_{r}n\rfloor })=({\sum }_{j=1}^r{\alpha }_j+o(1))n.$ For $\xi >0$ and sufficiently large $n$, let $G$ be a bipartite graph with bipartition $\{V_1,V_2\}$, $|V_1|=|V_2|=N$, where $N=(2+8\xi )n$. We prove that if $\delta (G)>(\frac{7}{8}+9\xi )N$, then any $2$-edge coloring of $G$ contains a monochromatic copy of $C_{2n}$.

19 pages

References in corpus (2)

• Three colour bipartite Ramsey number of cycles and paths
• Multicolour Ramsey numbers of paths and even cycles

Cited by in corpus (1)

• 3-colored asymmetric bipartite Ramsey number of connected matchings and cycles