Rainbow spanning trees in random edge-colored graphs

Abstract

A well known result of Erd\H{o}s and R\'enyi states that if $p = \frac{c \log n}{n}$ and $G$ is a random graph constructed from $G(n,p)$, $G$ is a.a.s. disconnected when $c 1$. When $c > 1$, we may equivalently say that $G$ a.a.s. contains a spanning tree. We find analogous thresholds in the setting of random edge-colored graphs. Specifically, we consider a family $\mathcal G$ of $n-1$ graphs on a common set $X$ of $n$ vertices, each of a different color, and each randomly chosen from $G(n,p)$, with $p = \frac{c \log n}{n^2}$. We show that when $c > 2$, there a.a.s. exists a spanning tree on $X$ using exactly one edge of each color, and we show that such a spanning tree a.a.s. does not exist when $c < 2$.Comment: It was discovered that the main result follows from Frieze, McKay, "Multicolored trees in random graphs," RSA, 199