Giant rainbow trees in sparse random graphs

Abstract

For any small constant $ϵ>0$, the Erdős-Rényi random graph $G(n,\frac{1+ϵ}{n})$ with high probability has a unique largest component which contains $(1\pm O\left(ϵ\right))2ϵn$ vertices. Let $G_c(n,p)$ be obtained by assigning each edge in $G(n,p)$ a color in $\left[c\right]$ independently and uniformly. Cooley, Do, Erde, and Missethan proved that for any fixed $\alpha >0$, $G_{\alpha n}(n,\frac{1+ϵ}{n})$ with high probability contains a rainbow tree (a tree that does not repeat colors) which covers $(1\pm O\left(ϵ\right))\frac{\alpha }{\alpha +1}ϵn$ vertices, and conjectured that there is one which covers $(1\pm O\left(ϵ\right))2ϵn$. In this paper, we achieve the correct leading constant and prove their conjecture correct up to a logarithmic factor in the error term, as we show that with high probability $G_{\alpha n}(n,\frac{1+ϵ}{n})$ contains a rainbow tree which covers $(1\pm O(ϵlog(1/ϵ)))2ϵn$ vertices.