# Covering graphs by monochromatic trees and Helly-type results for hypergraphs

## TL;DR

This paper connects the problem of covering graphs with monochromatic trees to a Helly-type hypergraph covering problem, providing new bounds and answers to longstanding questions in graph theory.

## Contribution

It establishes a novel link between graph covering problems and hypergraph Helly-type questions, offering new bounds and solutions for covering graphs with monochromatic trees.

## Key findings

- Derived accurate bounds for hypergraph covering problems.
- Provided new insights and answers to classical graph covering questions.
- Connected graph covering problems with hypergraph Helly-type properties.

## Abstract

How many monochromatic paths, cycles or general trees does one need to cover all vertices of a given $r$-edge-coloured graph $G$? 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 $H$ if it is known that any collection of a few edges of $H$ 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.