Covering random graphs with monochromatic trees

TL;DR

This paper investigates the minimum number of monochromatic connected components needed to cover vertices in random edge-coloured graphs, providing thresholds and solutions for the 3-colour case and general cases, connecting to hypergraph Helly-type problems.

Contribution

It introduces a modified hypergraph problem that precisely controls the covering problem in random graphs, resolving the 3-colour case and approximating the general case.

Findings

Identifies a threshold at $(rac{ ext{log } n}{n})^{1/4}$ for 3-colour graphs.

Provides an essentially complete solution for the 3-colour case.

Determines the approximate answer for general $r$-colour graphs near the boundedness threshold.

Abstract

Given an $r$-edge-coloured complete graph $K_n$, how many monochromatic connected components does one need in order to cover its vertex set? This natural question is a well-known essentially equivalent formulation of the classical Ryser's conjecture which, despite a lot of attention over the last 50 years, still remains open. A number of recent papers consider a sparse random analogue of this question, asking for the minimum number of monochromatic components needed to cover the vertex set of an $r$-edge-coloured random graph $\mathcal{G}(n,p)$. Recently, Buci\'{c}, Kor\'{a}ndi and Sudakov established a connection between this problem and a certain Helly-type local to global question for hypergraphs raised about 30 years ago by Erd\H{o}s, Hajnal and Tuza. We identify a modified version of the hypergraph problem which controls the answer to the problem of covering random graphs with monochromatic components more precisely. To showcase the power of our approach, we essentially resolve the $3$-colour case by showing that $(\log n / n)^{1/4}$ is a threshold at which point three monochromatic components are needed to cover all vertices of a $3$-edge-coloured random graph, answering a question posed by Kohayakawa, Mendon\c{c}a, Mota and Sch\"ulke. Our approach also allows us to determine the answer in the general $r$-edge coloured instance of the problem, up to lower order terms, around the point when it first becomes bounded, answering a question of Buci\'c, Kor\'andi and Sudakov.