# Monochromatic connected matchings in 2‐edge‐colored multipartite graphs

@article{Balogh2019MonochromaticCM, title={Monochromatic connected matchings in 2‐edge‐colored multipartite graphs}, author={J{\'o}zsef Balogh and Alexandr V. Kostochka and Mikhail Lavrov and Xujun Liu}, journal={Journal of Graph Theory}, year={2019}, volume={100}, pages={578 - 607} }

A matching M $M$ in a graph G $G$ is connected if all the edges of M $M$ are in the same component of G $G$ . 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…

## 4 Citations

### An improvement on Łuczak's connected matchings method

- MathematicsBulletin of the London Mathematical Society
- 2022

A connected matching in a graph G$G$ is a matching contained in a connected component of G$G$ . A well‐known method due to Łuczak reduces problems about monochromatic paths and cycles in complete…

### Monochromatic connected matchings in almost complete graphs

- Mathematics
- 2020

A connected matching in a graph $G$ is a matching that is contained in a connected component of $G$. A well-known method due to Łuczak reduces problems about monochromatic paths and cycles in…

### New Lower Bounds on the Size-Ramsey Number of a Path

- MathematicsElectron. J. Comb.
- 2022

We prove that for all graphs with at most $(3.75-o(1))n$ edges there exists a 2-coloring of the edges such that every monochromatic path has order less than $n$. This was previously known to be true…

### New lower bounds on the size-Ramsey number of a path. (English)

- Mathematics
- 2022

Electron. Summary: We prove that for all graphs with at most (3 . 75 − o (1)) n edges there exists a 2-coloring of the edges such that every monochromatic path has order less than n . This was…

## References

SHOWING 1-10 OF 29 REFERENCES

### Monochromatic connected matchings in almost complete graphs

- Mathematics
- 2020

A connected matching in a graph $G$ is a matching that is contained in a connected component of $G$. A well-known method due to Łuczak reduces problems about monochromatic paths and cycles in…

### 3‐Color bipartite Ramsey number of cycles and paths

- MathematicsJ. Graph Theory
- 2019

This paper determines asymptotically the $3$-colour bipartite Ramsey number of paths and (even) cycles.

### Long monochromatic paths and cycles in 2-edge-colored multipartite graphs

- MathematicsMoscow Journal of Combinatorics and Number Theory
- 2020

We solve four similar problems: For every fixed $s$ and large $n$, we describe all values of $n_1,\ldots,n_s$ such that for every $2$-edge-coloring of the complete $s$-partite graph…

### Monochromatic Cycles in 2-Coloured Graphs

- MathematicsCombinatorics, Probability and Computing
- 2012

Li, Nikiforov and Schelp [13] conjectured that any 2-edge coloured graph G with order n and minimum degree δ(G) > 3n/4 contains a monochromatic cycle of length ℓ, for all ℓ ∈ [4, ⌈n/2⌉]. We prove…

### Tripartite Ramsey numbers for paths

- MathematicsJ. Graph Theory
- 2007

It is shown that in any two-coloring of the edges of the complete tripartite graph K(n, n, n) there is a monochromatic path of length (1 − o(1)2n) since R(P2n +1, P2n+1) = 3n.

### Three colour bipartite Ramsey number of cycles and paths

- Mathematics
- 2018

The k-colour bipartite Ramsey number of a bipartite graph H is the least integer n for which every k-edge-coloured complete bipartite graph Kn,n contains a monochromatic copy of H. The study of…

### Multicolour Bipartite Ramsey Number of Paths

- MathematicsElectron. J. Comb.
- 2019

This paper determines asymptotically the $4$-colour bipartite Ramsey number of paths and cycles which are close to being tight and provides new upper bounds on the $k$- Colour bipartites Ramsey numbers which are Close to being Tight.

### Improved Bounds on the Multicolor Ramsey Numbers of Paths and Even Cycles

- MathematicsElectron. J. Comb.
- 2019

The upper bound of R_k(P_n) is improved to $(k - \frac{1}{2}+ o(1))n) and structural insights in connected graphs without a large matching are used.