Covering random graphs with monochromatic trees
Domagoj Brada\v{c}, Matija Buci\'c

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 -edge-coloured complete graph , 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 -edge-coloured random graph . 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…
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