Covering graphs by monochromatic trees and Helly-type results for hypergraphs
Abstract
How many monochromatic paths, cycles or general trees does one need to cover all vertices of a given -edge-coloured graph ? These problems were introduced in the 1960s and were intensively studied by various researchers over the last 50 years. In this paper, we establish a connection between this problem and the following natural Helly-type question in hypergraphs. Roughly speaking, this question asks for the maximum number of vertices needed to cover all the edges of a hypergraph if it is known that any collection of a few edges of has a small cover. We obtain quite accurate bounds for the hypergraph problem and use them to give some unexpected answers to several questions about covering graphs by monochromatic trees raised and studied by Bal and DeBiasio, Kohayakawa, Mota and Schacht, Lang and Lo, and Gir\~ao, Letzter and Sahasrabudhe.
Cite
@article{arxiv.1902.05055,
title = {Covering graphs by monochromatic trees and Helly-type results for hypergraphs},
author = {Matija Bucić and Dániel Korándi and Benny Sudakov},
journal= {arXiv preprint arXiv:1902.05055},
year = {2020}
}
Comments
20 pages including references plus 2 pages of an Appendix