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We consider the following random model for edge-colored graphs. A graph $G$on $n$ vertices is fixed, and a random subgraph $G_p$ is chosen by letting eachedge of $G$ remain independently with probability $p$. Then, each edge of $G_p$is colored uniformly at random from the set $[n-1]$. A result of Frieze andMcKay (Random Structures and Algorithms, 1994) implies that when $G = K_n$ and$p = (2 + \epsilon) \frac{\log n}{n}$ for some constant $\epsilon > 0$, then$G_p$ almost surely contains a rainbow spanning tree. In this paper, we showthat if $G$ is a $d$-regular $\Omega(n)$-edge-connected graph, then when $p =(2 + \epsilon) \frac {\log n}{d}$ for some constant $\epsilon > 0$, $G_p$almost surely contains a rainbow spanning tree. Our main tool is a newedge-replacement method for rainbow forests.

Rainbow spanning trees in random subgraphs of dense regular graphs