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The bipartite Ramsey numbers $BR(C_8, C_{2n})$

Mostafa Gholami, Yaser Rowshan

Abstract

For the given bipartite graphs $G_1,G_2,\ldots,G_t$, the multicolor bipartite Ramsey number $BR(G_1,G_2,\ldots,G_t)$ is the smallest positive integer $b$ such that any $t$-edge-coloring of $K_{b,b}$ contains a monochromatic subgraph isomorphic to $G_i$, colored with the $i$th color for some $1\leq i\leq t$. We compute the exact values of the bipartite Ramsey numbers $BR(C_8,C_{2n})$ for $n\geq2$.

The bipartite Ramsey numbers $BR(C_8, C_{2n})$

Abstract

For the given bipartite graphs , the multicolor bipartite Ramsey number is the smallest positive integer such that any -edge-coloring of contains a monochromatic subgraph isomorphic to , colored with the th color for some . We compute the exact values of the bipartite Ramsey numbers for .

Paper Structure

This paper contains 3 sections, 12 theorems, 14 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Some basic results
  3. Proof of the main results

Key Result

Theorem 1.1

For any $n\geq 2$, we have:

Figures (9)

  • Figure 1: Edge disjoint subgraphs $G$ and $\overline{G}$ of $K_{7,7}$
  • Figure 2: $x_3y_3\in E(C')$
  • Figure 3: $|\{x_3,y_3\}\cap V(C')|=1$$(x_3\in V(C'))$
  • Figure 4: $x=x_3, y=y_4$ and $x_3y_4\notin E(C')$
  • Figure 5: $x_3,x_4 \in X_1',y_4\in Y_1'$ and $x_3y_4\notin E(C')$
  • ...and 4 more figures

Theorems & Definitions (24)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • Theorem 2.4
  • Proposition 2.5
  • proof
  • Theorem 3.1
  • ...and 14 more