# Large Monochromatic Components in Edge Colorings of Graphs: A Survey

@inproceedings{Gyrfs2011LargeMC, title={Large Monochromatic Components in Edge Colorings of Graphs: A Survey}, author={Andr{\'a}s Gy{\'a}rf{\'a}s}, year={2011} }

The aim of this survey is to summarize an area of combinatorics that lies on the border of several areas: Ramsey theory, resolvable block designs, factorizations, fractional matchings and coverings, and partition covers.

## 40 Citations

Ramsey-Type Problems for Geometric Graphs

- Mathematics
- 2013

We survey some results and collect a set of open problems related to graph Ramsey theory with geometric constraints.

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…

Partitioning a graph into monochromatic connected subgraphs

- MathematicsJ. Graph Theory
- 2019

A variant of this problem, where the complete graph is replaced by a graph with large minimum degree, is considered, and two conjectures of Bal and DeBiasio are proved, for two and three colours.

Monochromatic diameter-2 components in edge colorings of the complete graph

- MathematicsInvolve, a Journal of Mathematics
- 2021

Gyarfas conjectured that in every r -edge-coloring of the complete graph K n there is a monochromatic component on at least n ∕ ( r − 1 ) vertices which has diameter at most 3. We show that for r = 3…

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…

Large components in r‐edge‐colorings of Kn have diameter at most five

- MathematicsJ. Graph Theory
- 2012

It is shown in this note that every r-edge-coloring of Kn contains a monochromatic component of diameter at most five on at least n/(r−1) vertices.

Covering complete graphs by monochromatically bounded sets

- MathematicsApplicable Analysis and Discrete Mathematics
- 2019

Given a k-colouring of the edges of the complete graph Kn, are there k-1 monochromatic components that cover its vertices? This important special case of the well-known Lov?sz-Ryser conjecture is…

Monochromatic components with many edges

- Mathematics
- 2022

. Given an r -edge-coloring of the complete graph K n , what is the largest number of edges in a monochromatic connected component? This natural question has only recently received the attention it…

Large monochromatic components of small diameter

- MathematicsJ. Graph Theory
- 2022

This note improves the result in the case of r = 3 and shows that in every 3-edge-coloring of Kn either there is a monochromatic component of diameter at most three on at least n/2 vertices or every color class is spanning and has diameter at least four.

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