Let $G=(V,E)$ be a graph. A (proper) $k$-edge-coloring is a coloring of the edges of $G$ such that any pair of edges sharing an endpoint receive distinct colors. A classical result of Vizing ensures that any simple graph $G$ admits a $(\Delta(G)+1)$-edge coloring where $\Delta(G)$ denotes the maximum degreee of $G$. Recently, Cabello raised the following question: given two graphs $G_1,G_2$ of maximum degree $\Delta$ on the same set of vertices $V$, is it possible to edge-color their (edge) union with $\Delta+2$ colors in such a way the restriction of $G$ to respectively the edges of $G_1$ and the edges of $G_2$ are edge-colorings? More generally, given $\ell$ graphs, how many colors do we need to color their union in such a way the restriction of the coloring to each graph is proper? In this short note, we prove that we can always color the union of the graphs $G_1,\ldots,G_\ell$ of maximum degree $\Delta$ with $\Omega(\sqrt{\ell} \cdot \Delta)$ colors and that there exist graphs for which this bound is tight up to a constant multiplicative factor. Moreover, for two graphs, we prove that at most $\frac 32 \Delta +4$ colors are enough which is, as far as we know, the best known upper bound.

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关于同时边缘着色的说明

令 $G=(V,E)$ 是一个图。一个（正确的）$k$-edge-coloring 是对$G$ 的边进行着色，这样共享一个端点的任何对边都会收到不同的颜色。Vizing 的经典结果确保任何简单的图 $G$ 承认 $(\Delta(G)+1)$-edge 着色，其中 $\Delta(G)$ 表示 $G$ 的最大度数。最近，Cabello 提出了以下问题：给定在同一组顶点 $V$ 上最大度数 $\Delta$ 的两个图 $G_1,G_2$，是否可以用 $\Delta+ 对它们的（边）并集进行边着色2$ 颜色以这种方式 $G$ 分别限制在 $G_1$ 的边缘和 $G_2$ 的边缘是边缘着色？更一般地说，给定 $\ell$ 图形，我们需要多少种颜色来为它们的并集着色，以使对每个图形的着色限制是适当的？在这个简短的说明中，我们证明我们总是可以用 $\Omega(\sqrt{\ell} \cdot \Delta)$ 颜色为最大度数 $\Delta$ 的图 $G_1,\ldots,G_\ell$ 的并集着色，并且有存在此边界紧至恒定乘法因子的图。此外，对于两个图，我们证明最多 $\frac 32 \Delta +4$ 颜色就足够了，据我们所知，这是最著名的上限。