Rainbow spanning trees in random subgraphs of dense regular graphs

TL;DR

This paper proves that in dense regular graphs, a random subgraph with edges randomly colored almost surely contains a rainbow spanning tree, extending known results from complete graphs to broader classes.

Contribution

It introduces a new edge-replacement method for rainbow forests and generalizes the existence of rainbow spanning trees to dense regular graphs.

Findings

Rainbow spanning trees exist in random subgraphs of dense regular graphs.

The new edge-replacement method is effective for rainbow forest constructions.

Results extend previous work from complete graphs to regular graphs with many edges.

Abstract

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 each edge 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 and McKay (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 show that 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 new edge-replacement method for rainbow forests.