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arXiv · 2102.12012

Rainbow spanning trees in random subgraphs of dense regular graphs

Peter Bradshaw
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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.

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Peter Bradshaw. 2021-02-24. Rainbow spanning trees in random subgraphs of dense regular graphs. https://arxiv.org/abs/2102.12012

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