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

@article{Ruszink2021MonochromaticDC,
title={Monochromatic diameter-2 components in edge colorings of the complete graph},
author={Mikl{\'o}s Ruszink{\'o} and Lang Song and Daniel P. Szabo},
journal={Involve, a Journal of Mathematics},
year={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 , 4 , 5 and 6 a diameter of 3 is best possible in this conjecture, constructing colorings where every monochromatic diameter-2 subgraph has strictly less than n ∕ ( r − 1 ) vertices.

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.Expand

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.Expand

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.Expand

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,… Expand

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.Expand

This paper is primarily concerned with a special case of one of the leading problems of mathematical logic, the problem of finding a regular procedure to determine the truth or falsity of any given… Expand