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A bound for the distinguishing index of regular graphs
European Journal of Combinatorics ( IF 1.0 ) Pub Date : 2020-05-22 , DOI: 10.1016/j.ejc.2020.103145
Florian Lehner , Monika Pilśniak , Marcin Stawiski

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 K2 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 κ such that every connected κ-regular graph has a distinguishing edge-colouring with two colours.



中文翻译:

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ķ2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κ 这样每个连接 κ-正则图具有两个不同的边缘色。

更新日期:2020-05-22
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