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arXiv 2003.05637math.COcs.DM
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有界度图闭邻域的无着色

Conflict-free coloring on closed neighborhoods of bounded degree graphs

Sriram Bhyravarapu, Subrahmanyam Kalyanasundaram, Rogers Mathew

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4 pages
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摘要：

图$G$的闭邻域无冲突色数，记为$χ_{CN}(G)$，是对图$G$的顶点进行着色所需的最小颜色数，使得对于每个顶点，在其闭邻域中存在恰好出现一次的颜色。Pach和Tardos [Combin.可能吧Comput. 2009]表明，对于任何$\varepsilon > 0$，$χ_{CN}(G) = O(\log^{2+\varepsilon} Δ)$，其中$Δ$是最大程度。在[Combin.可能吧Comput. 2014]，Glebov，Szabó和Tardos证明了图$G$与$χ_{CN}(G) = Ω(\log^2Δ)$的存在性。在本文中，我们通过证明$χ_{CN}(G) = O(\log^2 Δ)$。

英文摘要：

The closed neighborhood conflict-free chromatic number of a graph $G$, denoted by $χ_{CN}(G)$, is the minimum number of colors required to color the vertices of $G$ such that for every vertex, there is a color that appears exactly once in its closed neighborhood. Pach and Tardos [Combin. Probab. Comput. 2009] showed that $χ_{CN}(G) = O(\log^{2+\varepsilon} Δ)$, for any $\varepsilon > 0$, where $Δ$ is the maximum degree. In [Combin. Probab. Comput. 2014], Glebov, Szabó and Tardos showed existence of graphs $G$ with $χ_{CN}(G) = Ω(\log^2Δ)$. In this paper, we bridge the gap between the two bounds by showing that $χ_{CN}(G) = O(\log^2 Δ)$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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