# Ramsey‐type results for Gallai colorings

@article{Gyrfs2010RamseytypeRF, title={Ramsey‐type results for Gallai colorings}, author={Andr{\'a}s Gy{\'a}rf{\'a}s and G{\'a}bor N. S{\'a}rk{\"o}zy and Andr{\'a}s Seb{\"o} and Stanley M. Selkow}, journal={Journal of Graph Theory}, year={2010}, volume={64} }

A Gallai‐coloring of a complete graph is an edge coloring such that no triangle is colored with three distinct colors. Gallai‐colorings occur in various contexts such as the theory of partially ordered sets (in Gallai's original paper) or information theory. Gallai‐colorings extend 2‐colorings of the edges of complete graphs. They actually turn out to be close to 2‐colorings—without being trivial extensions. Here, we give a method to extend some results on 2‐colorings to Gallai‐colorings, among…

## 48 Citations

Disconnected Colors in Generalized Gallai‐Colorings

- MathematicsJ. Graph Theory
- 2013

It is shown that Gallai's property for two infinite families and it also holds when F is a path with at most six vertices, namely, they can be made bipartite by the removal of at most one edge.

Note on a generalization of Gallai-Ramsey numbers

- Mathematics
- 2019

A colored complete graph is said to be Gallai-colored if it contains no rainbow triangle. This property has been shown to be equivalent to the existence of a partition of the vertices (of every…

All partitions have small parts - Gallai-Ramsey numbers of bipartite graphs

- MathematicsDiscret. Appl. Math.
- 2019

Generalized Colorings of Graphs

- Mathematics
- 2016

A graph coloring is an assignment of labels called “colors” to certain elements of a graph subject to certain constraints. The proper vertex coloring is the most common type of graph coloring, where…

Gallai colorings and domination in multipartite digraphs

- MathematicsJ. Graph Theory
- 2012

It is said that there exists a h = h(β(D)) such that D has a dominating set of size at most h, applied to settle a problem related to generalized Gallai colorings, edge colorings of graphs without 3-colored triangles.

Gallai-Ramsey Numbers for Rainbow S3+ S_3^ + and Monochromatic Paths

- MathematicsDiscuss. Math. Graph Theory
- 2022

Abstract Motivated by Ramsey theory and other rainbow-coloring-related problems, we consider edge-colorings of complete graphs without rainbow copy of some fixed subgraphs. Given two graphs G and H,…

## References

SHOWING 1-10 OF 46 REFERENCES

Size of Monochromatic Double Stars in Edge Colorings

- MathematicsGraphs Comb.
- 2008

It is shown that in every r-coloring of the edges of Kn there is a monochromatic double star with at least n(r+1)+r-1}{r^2+1} vertices, which improves a bound of Mubayi for the largest monochromeatic subgraph of diameter at most three.

Edge colorings of complete graphs without tricolored triangles

- MathematicsJ. Graph Theory
- 2004

We show some consequences of results of Gallai concerning edge colorings of complete graphs that contain no tricolored triangles. We prove two conjectures of Bialostocki and Voxman about the…

Finding Large p-Colored Diameter Two Subgraphs

- MathematicsGraphs Comb.
- 1999

It is shown for k≥1 and k\2≤p≤k that there is always a p-colored diameter two subgraph of Kn containing at least vertices and that this is best possible up to an additive constant l satisfying 0≤l.

Domination in colored complete graphs

- MathematicsJ. Graph Theory
- 1989

There exists X c V(K,) such that 1x1 I t and X monochromatically dominates all but at most n/2’ vertices of K, and X can be constructed by a fast greedy algorithm.

Edge-colored complete graphs with precisely colored subgraphs

- MathematicsComb.
- 1983

The main object of this paper is to describe the behavior of the functionf(s,t;k), usually thinking ofs andt fixed, and lettingk become large.

Lambda composition

- MathematicsJ. Graph Theory
- 1997

The decomposition of a graph into its lambda subgraphs is described and this is used to prove the decomposition theorem of Gallai.

Recent results on generalized Ramsey theory for graphs

- Mathematics
- 1972

Virtually all of the known results on generalized Ramsey theory for graphs have been reported here, and the most general method of proof was brute force. There is certainly a need for more powerful…

Generalizing the Ramsey Problem through Diameter

- MathematicsElectron. J. Comb.
- 2002

The results include determining $f_1^k(K_n)$, which is equivalent to determining classical Ramsey numbers for multicolorings, and a construction due to Calkin implies that $f-3^k (K-n) \le {{n}\over {k-1}} + k-1$ when $k- 1$ is a prime power.

A note on perfect graphs

- Mathematics
- 1986

Let the lines of a complete graph be 3-colored so that no triangle gets 3 different colors. If two of these colors form perfect graphs then so does the third.