Given a multigraph, suppose that each vertex is given a local assignment of $k$ colours to its incident edges. We are interested in whether there is a choice of one local colour per vertex such that no edge has both of its local colours chosen. The least $k$ for which this is always possible given any set of local assignments we call the {\em single-conflict chromatic number} of the graph. This parameter is closely related to separation choosability and adaptable choosability. We show that single-conflict chromatic number of simple graphs embeddable on a surface of Euler genus $g$ is $O(g^{1/4}\log g)$ as $g\to\infty$. This is sharp up to the logarithmic factor.

中文翻译：

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单一冲突着色


给定一个多重图，假设每个顶点都被赋予 $k$ 颜色到其关联边的局部分配。我们感兴趣的是每个顶点是否可以选择一种局部颜色，以便没有边同时选择两种局部颜色。给定任何一组局部分配，这总是可能的最小$k$，我们称之为图的{\em单冲突色数}。该参数与分离选择性和适应性选择性密切相关。我们证明了可嵌入欧拉属 $g$ 表面上的简单图的单冲突色数为 $O(g^{1/4}\log g)$ 作为 $g\to\infty$。这对于对数因子来说是尖锐的。