Monochromatic trees in random graphs

TL;DR

This paper determines the threshold for a Ramsey-type property in random graphs, where every 2-coloring results in two disjoint monochromatic trees partitioning all vertices.

Contribution

It establishes the exact threshold for the property in two-colorings, advancing understanding of monochromatic tree partitions in random graphs.

Findings

Threshold for the property in two-colorings is identified.

Provides a precise condition for monochromatic tree partitions.

Advances the theory of Ramsey properties in random graphs.

Abstract

Bal and DeBiasio [Partitioning random graphs into monochromatic components, Electron. J. Combin. 24 (2017), Paper 1.18] put forward a conjecture concerning the threshold for the following Ramsey-type property for graphs $G$: every $k$-colouring of the edge set of $G$ yields $k$ pairwise vertex disjoint monochromatic trees that partition the whole vertex set of $G$. We determine the threshold for this property for two colours.