Rainbow trees in uniformly edge‐colored graphs

Abstract

AbstractWe 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 , 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 et al. 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 uniform coloring of the randomly perturbed graph , using colors, where 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 , where is independent of , a.a.s. admits a copy of any given bounded‐degree spanning tree. Finally, and with as above, we prove that a uniform coloring of using 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.