A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back more than two hundred years to the work of Euler on Latin squares. Since then rainbow structures have been the focus of extensive research and have found applications in the areas of graph labelling and decomposition. An edge-colouring is locally $k$-bounded if each vertex is contained in at most $k$ edges of the same colour. In this paper we prove that any such edge-colouring of the complete graph $K_n$ contains a rainbow copy of every tree with at most $(1-o(1))n/k$ vertices. As a locally $k$-bounded edge-colouring of $K_n$ may have only $(n-1)/k$ distinct colours, this is essentially tight. As a corollary of this result we obtain asymptotic versions of two long-standing conjectures in graph theory. Firstly, we prove an asymptotic version of Ringel's conjecture from 1963, showing that any $n$-edge tree packs into the complete graph $K_{2n+o(n)}$ to cover all but $o(n^2)$ of its edges. Secondly, we show that all trees have an almost-harmonious labelling. The existence of such a labelling was conjectured by Graham and Sloane in 1980. We also discuss some additional applications.

中文翻译：

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将彩虹树嵌入到图形标记和分解中的应用中

如果边着色图的所有边都具有不同的颜色，则该子图称为彩虹。彩虹子图的研究可以追溯到两百多年前欧拉对拉丁方阵的研究。从那时起，彩虹结构一直是广泛研究的焦点，并在图标记和分解领域得到了应用。如果每个顶点最多包含在相同颜色的 $k$ 边中，则边着色是局部 $k$-bounded。在本文中，我们证明完整图 $K_n$ 的任何此类边缘着色都包含每棵树的彩虹副本，最多具有 $(1-o(1))n/k$ 个顶点。由于 $K_n$ 的局部 $k$-bounded 边缘着色可能只有 $(n-1)/k$ 不同的颜色，这基本上是紧的。作为这一结果的推论，我们获得了图论中两个长期存在的猜想的渐近版本。首先，我们证明了 1963 年林格尔猜想的渐近版本，表明任何 $n$-边树都打包到完整图 $K_{2n+o(n)}$ 中以覆盖除 $o(n^2)$ 之外的所有图边缘。其次，我们证明所有的树都有一个几乎和谐的标签。Graham 和 Sloane 于 1980 年推测了这种标记的存在。我们还讨论了一些其他应用。