[Paper Review] Bipartite Ramsey numbers of large cycles

Shaoqiang Liu, Yuejian Peng|arXiv (Cornell University)|Aug 30, 2018

Limits and Structures in Graph Theory18 references3 citations

TL;DR

This paper establishes asymptotically tight bounds for bipartite Ramsey numbers of large even cycles under various coloring conditions. It proves that for $ r \geq 2 $, the bipartite Ramsey number $ br(C_{2\lfloor \alpha_1 n \rfloor}, \dots, C_{2\lfloor \alpha_r n \rfloor}) $ is asymptotically $ (\sum \alpha_i + o(1))n $, with explicit degree conditions ensuring monochromatic cycles in multicolorings, using extremal graph theory and regularity-based arguments.

ABSTRACT

For an integer $r\geq 2$ and bipartite graphs $H_i$, where $1\leq i\leq r$, the bipartite Ramsey number $br(H_1,H_2,\ldots,H_r)$ 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 $i$, $1\leq i\leq r$. We show that for $α_1,α_2>0$, $br(C_{2\lfloor α_1 n floor},C_{2\lfloor α_2 n floor})=(α_1+α_2+o(1))n$. We also show that if $r\geq 3, α_1,α_2>0, α_{j+2}\geq [(j+2)!-1]\sum^{j+1}_{i=1} α_i$ for $j=1,2,\ldots,r-2$, then $br(C_{2\lfloor α_1 n floor},C_{2\lfloor α_2 n floor},\ldots,C_{2\lfloor α_r n floor})=(\sum^r_{j=1} α_j+o(1))n.$ For $ξ>0$ and sufficiently large $n$, let $G$ be a bipartite graph with bipartition $\{V_1,V_2\}$, $|V_1|=|V_2|=N$, where $N=(2+8ξ)n$. We prove that if $δ(G)>(\frac{7}{8}+9ξ)N$, then any $2$-edge coloring of $G$ contains a monochromatic copy of $C_{2n}$.

Motivation & Objective

• To determine the asymptotic behavior of bipartite Ramsey numbers for multiple large even cycles under multicolor edge coloring.
• To establish tight upper and lower bounds on $ br(C_{2\lfloor \alpha_1 n \rfloor}, \dots, C_{2\lfloor \alpha_r n \rfloor}) $ as $ n \to \infty $.
• To identify the minimum minimum-degree threshold in a balanced bipartite graph that guarantees a monochromatic even cycle under 2-coloring.
• To investigate the sharpness of the bound $ c = \frac{7}{8} $ in the context of Ramsey-Turán type problems for even cycles.
• To resolve the extremal threshold for monochromatic cycle containment in dense bipartite graphs via structural and regularity-based arguments.

Proposed method

• Utilizes the regularity lemma and sparse regularity techniques to analyze edge-colored bipartite graphs with high minimum degree.
• Applies a decomposition argument to partition vertex sets into structured parts (e.g., $ C, S, T, U $) to control neighborhood expansions.
• Employs connected matching constructions in subgraphs induced by selected vertex sets to embed large cycles.
• Uses degree conditions $ \delta(G) > (\frac{7}{8} + 9\xi)N $ to ensure sufficient connectivity and edge density for cycle containment.
• Applies extremal constructions (e.g., $ \widetilde{H} $) to demonstrate lower bounds on the required minimum degree $ c $, showing $ c \geq \frac{3}{4} $.
• Derives recursive inequalities on cycle sizes $ \alpha_{j+2} \geq [(j+2)! - 1] \sum_{i=1}^{j+1} \alpha_i $ to ensure multicolor Ramsey thresholds hold.

Experimental results

Research questions

• RQ1 What is the asymptotic value of the bipartite Ramsey number $ br(C_{2\lfloor \alpha_1 n \rfloor}, C_{2\lfloor \alpha_2 n \rfloor}) $ as $ n \to \infty $?
• RQ2 How does the bipartite Ramsey number scale for $ r \geq 3 $ large even cycles when cycle sizes grow linearly with $ n $?
• RQ3 What is the minimal minimum-degree threshold $ c $ such that any $ 2 $-edge-colored bipartite graph with $ \delta(G) \geq cN $ contains a monochromatic $ C_{2n} $?
• RQ4 Can the bound $ c = \frac{7}{8} $ be improved, or is $ c = \frac{3}{4} $ the true threshold for monochromatic cycle containment?
• RQ5 Under what structural conditions on the edge coloring and degree distribution does a large monochromatic cycle emerge in a bipartite graph?

Key findings

• 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 $, establishing tight asymptotic growth.
• For $ r \geq 3 $, under the condition $ \alpha_{j+2} \geq [(j+2)! - 1] \sum_{i=1}^{j+1} \alpha_i $, $ br(C_{2\lfloor \alpha_1 n \rfloor}, \dots, C_{2\lfloor \alpha_r n \rfloor}) = (\sum_{j=1}^r \alpha_j + o(1))n $.
• If $ G $ is a balanced bipartite graph with $ |V_1| = |V_2| = N = (2 + 8\xi)n $ and $ \delta(G) > (\frac{7}{8} + 9\xi)N $, then any 2-edge coloring of $ G $ contains a monochromatic $ C_{2n} $.
• The example $ \widetilde{H} $ with $ \delta(\widetilde{H}) = \frac{3}{4}N $ and no monochromatic $ C_{4n} $ shows that $ c \geq \frac{3}{4} $ is necessary, and the bound $ c = \frac{7}{8} $ is tight up to this threshold.
• Connected matchings in structured subgraphs (e.g., $ G_B[H_1, H_2] $) are used to embed large cycles, with degree conditions ensuring connectivity and expansion.
• The proof relies on iterative application of regularity and degree-distribution arguments to identify subgraphs with sufficient connectivity to support large monochromatic cycles.