Rainbow trees in uniformly edge-colored graphs

We obtain sufficient conditions for the emergence of spanning and almost-spanning bounded-degree rainbow trees in various host graphs, having their edges colored independently and uniformly at random, using a predetermined palette. Our first result asserts that a uniform coloring of (Formula presented.), using a palette of size (Formula presented.), a.a.s. admits a rainbow copy of any given bounded-degree tree on at most (Formula presented.) vertices, where (Formula presented.) is arbitrarily small yet fixed. This serves as a rainbow variant of a classical result by Alon et al. pertaining to the embedding of bounded-degree almost-spanning prescribed trees in (Formula presented.), where (Formula presented.) is independent of (Formula presented.). Given an (Formula presented.) -vertex graph (Formula presented.) with minimum degree at least (Formula presented.), where (Formula presented.) is fixed, we use our aforementioned result in order to prove that a uniform coloring of the randomly perturbed graph (Formula presented.), using (Formula presented.) colors, where (Formula presented.) is arbitrarily small yet fixed, a.a.s. admits a rainbow copy of any given bounded-degree spanning tree. This can be viewed as a rainbow variant of a result by Krivelevich et al. who proved that (Formula presented.), where (Formula presented.) is independent of (Formula presented.), a.a.s. admits a copy of any given bounded-degree spanning tree. Finally, and with (Formula presented.) as above, we prove that a uniform coloring of (Formula presented.) using (Formula presented.) colors a.a.s. admits a rainbow spanning tree. Put another way, the trivial lower bound on the size of the palette required for supporting a rainbow spanning tree is also sufficient, essentially as soon as the random perturbation a.a.s. has edges.