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

中文翻译：

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图的局部有界着色中的彩虹结构

我们研究的边缘色准随机图形近似分解成彩虹横跨结构：一个图形的边缘着色是本地 -bounded如果每个顶点入射到至多每种颜色的边缘，并且是（全球 ） -bounded 如果每个颜色出现大多数时候。我们的结果表明存在：（1）适当的边缘颜色近似分解为彩虹的近跨周期；（2）将边缘着色近似分解为彩虹哈密顿循环，条件是着色是有界的和局部有界的；和（3）近似分解为任何数组的完整横截面，前提是每个符号都出现总次数，每行或每列仅次数。除了对数因素，这些界限基本上是最好的。我们还证明了彩虹因子的类似物，其中有任何固定的图。（1）和（2）都暗示了Brualdi-Hollingsworth猜想关于彩虹分解成树的近似版本。