Covering $3$-edge-coloured random graphs with monochromatic trees

TL;DR

This paper proves that for sufficiently dense random graphs with three edge colours, three monochromatic trees can cover all vertices, improving previous bounds for such coverings.

Contribution

It establishes a new threshold for covering vertices with three monochromatic trees in 3-edge-coloured random graphs, refining earlier results.

Findings

For $p \,\gg\, n^{-1/6} (\ln n)^{1/6}$, three monochromatic trees cover all vertices

Improves previous bounds on monochromatic tree coverings in 3-coloured random graphs

Shows the existence of such coverings with high probability in dense random graphs

Abstract

We investigate the problem of determining how many monochromatic trees are necessary to cover the vertices of an edge-coloured random graph. More precisely, we show that for $p\gg n^{-1/6}{(\ln n)}^{1/6}$, in any $3$-edge-colouring of the random graph $G(n,p)$ we can find three monochromatic trees such that their union covers all vertices. This improves, for three colours, a result of Buci\'c, Kor\'andi and Sudakov.