[Paper Review] Monochromatic connected matchings in 2-edge-colored multipartite graphs

József Balogh, Alexandr Kostochka+2 more|arXiv (Cornell University)|May 12, 2019

Limits and Structures in Graph Theory16 references4 citations

TL;DR

This paper establishes exact Ramsey-type bounds for monochromatic connected matchings in 2-edge-colored complete multipartite graphs, proving that under certain size conditions on the partite sets, a monochromatic connected matching of size $ n $ is guaranteed. It also provides a stability theorem, showing that either such a large matching exists or the edge-coloring is highly structured, extending prior asymptotic results to exact thresholds.

ABSTRACT

A matching $M$ in a graph $G$ is connected if all the edges of $M$ are in the same component of $G$. Following Łuczak,there have been many results using the existence of large connected matchings in cluster graphs with respect to regular partitions of large graphs to show the existence of long paths and other structures in these graphs. We prove exact Ramsey-type bounds on the sizes of monochromatic connected matchings in $2$-edge-colored multipartite graphs. In addition, we prove a stability theorem for such matchings.

Motivation & Objective

• To determine exact conditions under which every 2-edge-coloring of a complete $ s $-partite graph $ K_{n_1, _2, _s} $ contains a monochromatic connected matching of size $ n $.
• To generalize and sharpen prior asymptotic results on monochromatic connected matchings in dense multipartite graphs.
• To establish a stability theorem characterizing when large monochromatic connected matchings must exist or when the edge-coloring must be highly structured.
• To provide exact Ramsey-type thresholds for long paths and cycles in 2-edge-colored complete multipartite graphs, particularly for $ K_{n,n,n} $.

Proposed method

• Use of regularity-based methods and connected matching arguments inspired by Łuczak’s work on cluster graphs.
• Application of the Szemerédi Regularity Lemma and analysis of connected matchings in cluster graphs to derive exact bounds.
• Introduction of a stability framework: if no large monochromatic connected matching exists, the coloring must be structurally constrained.
• Construction of a $(35\gamma, 1, 2)$-bad partition to analyze extremal cases and derive contradictions under size assumptions.
• Use of matching decomposition and edge-switching techniques to modify matchings and satisfy structural constraints.
• Combination of extremal graph theory with Ramsey-theoretic reasoning to prove exact thresholds.

Experimental results

Research questions

• RQ1 What is the exact minimum size of the partite sets in a complete $ s $-partite graph that guarantees a monochromatic connected matching of size $ n $ under any 2-edge-coloring?
• RQ2 Under what structural conditions on the edge-coloring of a 2-edge-colored complete multipartite graph can large monochromatic connected matchings be avoided?
• RQ3 How do the exact bounds for monochromatic connected matchings in $ K_{n,n,n} $ relate to the Ramsey numbers for long paths and cycles?
• RQ4 Can a stability theorem be established that characterizes the extremal colorings avoiding large monochromatic connected matchings?

Key findings

• The paper establishes that for $ K_{n_1, _2, _s} $ with $ n_1 \geq n_2 \geq \cdots \geq n_s $, a monochromatic connected matching of size $ n $ exists if $ N = \sum n_i \geq 3n - 1 $ and $ N - n_1 \geq 2n - 1 $, and these bounds are tight.
• A stability theorem is proven: if no monochromatic connected matching of size $ (1+\gamma)n $ exists, then the coloring must be highly structured, with a $ (35\gamma, 1, 2) $-bad partition existing.
• The exact bound $ K_{n,n,n} \mapsto (P_{2n+1}, P_{2n+1}) $ is established as a consequence, confirming a conjecture of Gyárfás, Ruszinkó, Sárközy, and Szemerédi.
• The results extend prior asymptotic bounds to exact thresholds, resolving open problems in Ramsey theory for paths and cycles in 2-edge-colored complete multipartite graphs.
• The paper provides a refined structural characterization of extremal colorings that avoid large monochromatic connected matchings, using partitioning and matching modification techniques.