Montgomery关于正则图动态着色猜想的一个反例
A counterexample to Montgomery's conjecture on dynamic colourings of regular graphs
摘要：
图的\emph{dynamic colouring}是图的一个真染色，其中非叶顶点的邻域都不是单色的。图$G$的\emph{dynamic colouring number} $χ_2(G)$是$G$的动态着色所需的最少颜色数。Montgomery证明了$χ_2(G) \leq χ(G) + 2$对所有正则图$G$都是有效的，这将显著提高当前最佳上界$χ_2(G) \leq 2χ(G)$ .然而，在本文中，我们证明了最后一个上界是尖锐的，通过构造，对于每一个整数$n \geq 2$，一个正则图$G$与$χ(G) = n$，但$χ_2(G) = 2n$。特别是，这推翻了蒙哥马利的猜想。
英文摘要：
A \emph{dynamic colouring} of a graph is a proper colouring in which no neighbourhood of a non-leaf vertex is monochromatic. The \emph{dynamic colouring number} $χ_2(G)$ of a graph $G$ is the least number of colours needed for a dynamic colouring of $G$. Montgomery conjectured that $χ_2(G) \leq χ(G) + 2$ for all regular graphs $G$, which would significantly improve the best current upper bound $χ_2(G) \leq 2χ(G)$. In this note, however, we show that this last upper bound is sharp by constructing, for every integer $n \geq 2$, a regular graph $G$ with $χ(G) = n$ but $χ_2(G) = 2n$. In particular, this disproves Montgomery's conjecture.
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。
