Rainbow structures in locally bounded colorings of graphs

We study approximate decompositions of edge‐colored quasirandom graphs into rainbow spanning structures: an edge‐coloring of a graph is locally 𝓁‐bounded if every vertex is incident to at most 𝓁 edges of each color, and is (globally) g‐bounded if every color appears at most g times. Our results imply the existence of: (1) approximate decompositions of properly edge‐colored 𝐾_n into rainbow almost‐spanning cycles; (2) approximate decompositions of edge‐colored 𝐾_n into rainbow Hamilton cycles, provided that the coloring is (1−o(1))n/2 ‐bounded and locally o(n/log⁴n) ‐bounded; and (3) an approximate decomposition into full transversals of any nxn array, provided each symbol appears (1−o(1))n times in total and only o(n/log²n) times in each row or column. Apart from the logarithmic factors, these bounds are essentially best possible. We also prove analogues for rainbow F‐factors, where F is any fixed graph. Both (1) and (2) imply approximate versions of the Brualdi‐Hollingsworth conjecture on decompositions into rainbow spanning trees.