Rainbow structures in locally bounded colorings of graphs

Abstract

We study approximate decompositions of edge-colored quasirandom graphs into rainbow spanning structures: an edge-coloring of a graph is locallyl-bounded if every vertex is incident to at most l edges of each color, and is (globally) g-boundedif every color appears at most g times. Our results imply the existence of: (1) approximate decompositions of properly edge-colored Kn into rainbow almost-spanning cycles; (2) approximate decompositions of edge-colored Kn into rainbow Hamilton cycles, provided that the coloring is (1-o(1))n2-bounded and locally o(nlog4n)-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(nlog2n) 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.