A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. Our main result implies that, given any optimal colouring of a sufficiently large complete graph $K_{2n}$, there exists a decomposition of $K_{2n}$ into isomorphic rainbow spanning trees. This settles conjectures of Brualdi–Hollingsworth (from 1996) and Constantine (from 2002) for large graphs.

中文翻译：

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分解为同构彩虹生成树

如果边缘着色图的所有子边缘都有不同的颜色，则该子图称为彩虹。我们的主要结果表明，给定足够大的完整图的最佳着色${ķ}_{2ñ}$，存在分解 ${ķ}_{2ñ}$变成同构彩虹，横跨树木。这解决了Brualdi-Hollingsworth（来自1996）和君士坦丁（来自2002）关于大图的猜想。