arXiv 1803.033162018-03math.CO

嵌入彩虹树及其在图标号和分解中的应用

Embedding rainbow trees with applications to graph labelling and decomposition

Richard Montgomery, Alexey Pokrovskiy, Benny Sudakov

备注：: 22 pages

摘要：

边着色图的一个子图称为彩虹，如果它的所有边都有不同的颜色。彩虹子图的研究可以追溯到两百多年前欧拉关于拉丁方的工作。从那时起，彩虹结构一直是广泛研究的焦点，并在图的标记和分解领域中找到了应用。一个边着色是局部$k$ -有界的，如果每个顶点至多包含在相同颜色的$k$边中.本文证明了完全图$K_n$的任何边着色都包含至多有$(1-o(1))n/k$顶点的树的彩虹拷贝。由于$K_n$的局部$k$有界边着色可能仅具有$(n-1)/k$不同的颜色，因此这本质上是紧密的。作为这个结果的一个推论，我们得到了图论中两个长期存在的定理的渐近版本。首先，我们证明了1963年Ringel猜想的一个渐近版本，证明了任何$n$ -边树都可以填充到完全图$K_{2n+o(n)}$中，以覆盖除$o(n^2)$之外的所有边。其次，我们证明了所有的树都有一个几乎和谐的标签。1980年，格雷厄姆和斯隆证实了这种标签的存在。我们还讨论了一些其他的应用。

英文摘要：

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.

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英文摘要：

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