# Size of Monochromatic Double Stars in Edge Colorings

@article{Gyrfs2008SizeOM,
title={Size of Monochromatic Double Stars in Edge Colorings},
author={Andr{\'a}s Gy{\'a}rf{\'a}s and G{\'a}bor N. S{\'a}rk{\"o}zy},
journal={Graphs and Combinatorics},
year={2008},
volume={24},
pages={531-536}
}
• Published 1 November 2008
• Mathematics, Computer Science
• Graphs and Combinatorics
We show that in every r-coloring of the edges of Kn there is a monochromatic double star with at least $$\frac{n(r+1)+r-1}{r^2}$$ vertices. This result is sharp in asymptotic for r = 2 and for r≥ 3 improves a bound of Mubayi for the largest monochromatic subgraph of diameter at most three. When r-colorings are replaced by local r-colorings, our bound is $$\frac{n(r+1)+r-1}{r^2+1}$$.
14 Citations

## Topics from this paper

Note on 2-edge-colorings of complete graphs with small monochromatic k-connected subgraphs
• Mathematics
• 2014
Bollobás and Gyárfás conjectured that for n > 4 (k − 1) every 2-edge-coloring of Kn contains a monochromatic k-connected subgraph with at least n − 2k + 2 vertices. Liu, et al. proved that the
Monochromatic 4-connected subgraphs in constrained 2-edge-colorings of Kn
• Mathematics
• 2012
Bollobas and Gyarfas conjectured that for n >= 4k-3 every 2-edge-coloring of Kn contains a monochromatic k-connected subgraph with at least n -2k + 2 vertices. It was proved that the conjecture holds
Large components in r-edge-colorings of Kn have diameter at most five
• M. Ruszinkó
• Mathematics, Computer Science
J. 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.
Large monochromatic components of small diameter
• Computer Science, Mathematics
J. 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.
Monochromatic homeomorphically irreducible trees in $2$-edge-colored complete graphs
• Mathematics
• 2018
It has been known that every 2-edge-colored complete graph has a monochromatic connected spanning subgraph. In this paper, we study a condition which can be imposed on such monochromatic subgraph,
Ramsey-type results for Gallai colorings
• Computer Science
J. Graph Theory
• 2010
It is proved that in every G-coloring of Kn there exists each of the following: a monochromatic double star with at least 3n+1 4 vertices; and RG(r,K3) can be determined exactly.
Ramsey-type results for Gallai colorings
• Mathematics
• 2010
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
Large Monochromatic Triple Stars in Edge Colourings
It is proved that for every r-edge-colouring of Kn there is a monochromatic triple star of order at least n/r-1, improving Ruszinko's result 2012.
Monochromatic Structures in Edge-coloured Graphs and Hypergraphs - A survey
• Mathematics
• 2015
Given a graph whose edges are coloured, on how many vertices can we find a monochromatic subgraph of a certain type, such as a connected subgraph, or a cycle, or some type of tree? Also, how many
Large Monochromatic Components in Edge Colorings of Graphs: A Survey
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,

## References

SHOWING 1-10 OF 32 REFERENCES
Highly connected monochromatic subgraphs
• Computer Science, Mathematics
Discret. Math.
• 2008
Domination in colored complete graphs
• Mathematics, Computer Science
J. 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.
Size of monochromatic components in local edge colorings
• Computer Science, Mathematics
Discret. Math.
• 2008
Edge colorings of complete graphs without tricolored triangles
• Computer Science, Mathematics
J. 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
Generalized ramsey theory for graphs, x: double stars
• Computer Science, Mathematics
Discret. Math.
• 1979
Finding Large p-Colored Diameter Two Subgraphs
• Mathematics, Computer Science
Graphs 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.
Highly connected monochromatic subgraphs of multicolored graphs
• Mathematics
• 2009
We consider the following question of Bollobas: given an r-coloring of E(Kn), how large a k-connected subgraph can we find using at most s colors? We provide a partial solution to this problem when
Ramsey numbers for local colorings
• Mathematics, Computer Science
Graphs Comb.
• 1987
The localk-Ramsey numberrlock(G) is a natural generalization of the usual Ramsey numberrk(G), defined for usualk-colorings, and the results reflect the relationship betweenrk (G) and rlock( G) for certain classes of graphs.
Highly connected multicoloured subgraphs of multicoloured graphs
• Computer Science, Mathematics
Discret. Math.
• 2008
Generalized Ramsey theory for graphs
• Mathematics
• 1972
The classical Ramsey numbers [7] involve the occurrence of monochromatic complete subgraphs in line-colored complete graphs. By removing the completeness requirements and admitting arbitrary