In 2018, Bai, Fujita and Zhang [Discrete Math. 341 (2018), no. 6, 1523–1533] introduced the concept of a kernel by rainbow paths (for short, RP-kernel) of an arc-coloured digraph D D , which is a subset S S of vertices of D D such that ( a a ) there exists no rainbow path for any pair of distinct vertices of S S , and ( b b ) every vertex outside S S can reach S S by a rainbow path in D D . They showed that it is NP-hard to recognize whether an arc-coloured digraph has an RP-kernel and it is NP-complete to decide whether an arc-coloured tournament has an RP-kernel. In this paper, we give the sufficient conditions for the existence of an RP-kernel in arc-coloured unicyclic digraphs, semicomplete digraphs, quasi-transitive digraphs and bipartite tournaments, and prove that these arc-coloured digraphs have RP-kernels if certain “short” cycles and certain “small” induced subdigraphs are rainbow.

中文翻译：

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在弧形有向图中的彩虹路径的核上

在2018年，Bai，Fujita和Zhang [Discrete Math。341（2018），否 [6，1523–1533]引入了由弧形有色图DD的彩虹路径（简称RP核）组成的核的概念，该路径是DD顶点的子集SS，因此（aa）不存在彩虹路径对于SS的任意一对不同的顶点，（bb）SS外部的每个顶点都可以通过DD中的一条彩虹路径到达SS。他们表明，识别圆弧形有向图是否具有RP内核是NP难的，而决定圆弧形锦标赛是否具有RP内核是NP完整的。在本文中，我们给出了圆弧色单圈图，半完全图，拟传递图和二分对立比赛中RP核存在的充分条件，