Rainbow trees in uniformly edge-coloured graphs
Elad Aigner-Horev, Dan Hefetz, Abhiruk Lahiri

TL;DR
This paper establishes conditions under which uniformly edge-coloured random and perturbed graphs contain rainbow spanning and almost-spanning trees with bounded degree, extending classical embedding results to rainbow variants.
Contribution
It introduces new probabilistic thresholds for rainbow spanning trees in random and perturbed graphs, generalizing classical embedding theorems to rainbow settings.
Findings
Rainbow almost-spanning trees exist in G(n, C/n) with high probability.
Rainbow spanning trees are present in perturbed graphs with slightly more than n colours.
A rainbow spanning tree exists with n-1 colours in the union of a fixed graph and a random graph.
Abstract
We obtain sufficient conditions for the emergence of spanning and almost-spanning bounded-degree {\sl rainbow} trees in various host graphs, having their edges coloured independently and uniformly at random, using a predetermined palette. Our first result asserts that a uniform colouring of , using a palette of size , a.a.s. admits a rainbow copy of any given bounded-degree tree on at most vertices, where is arbitrarily small yet fixed. This serves as a rainbow variant of a classical result by Alon, Krivelevich, and Sudakov pertaining to the embedding of bounded-degree almost-spanning prescribed trees in , where is independent of . Given an -vertex graph with minimum degree at least , where is fixed, we use our aforementioned result in order to prove that a…
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Taxonomy
TopicsStochastic processes and statistical mechanics · Limits and Structures in Graph Theory · Advanced Graph Theory Research
