An edge-colouring of a graph is distinguishing if the only automorphism that preserves the colouring is the identity. It has been conjectured that all but finitely many connected, finite, regular graphs admit a distinguishing edge-colouring with two colours. We show that all such graphs except $K_2$ admit a distinguishing edge-colouring with three colours. This result also extends to infinite, locally finite graphs. Furthermore, we are able to show that there are arbitrary large infinite cardinals $\kappa$ such that every connected $\kappa$-regular graph has a distinguishing edge-colouring with two colours.

中文翻译：

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正则图的区别索引的界

如果唯一保留颜色的自同构性是同一性，则可以区分图的边缘颜色。据推测，除了有限数量外，所有相连的，有限的，规则的图都允许使用两种颜色区分边缘。我们证明了所有这些图除了${ķ}_2$承认三种不同的边缘色。此结果还扩展到无限图，局部有限图。此外，我们能够证明有任意大的无限基数$\kappa$ 这样每个连接 $\kappa$-正则图具有两个不同的边缘色。