颜色细化的迭代次数
The Iteration Number of Colour Refinement
摘要：
颜色细化过程及其推广到更高的维度，Weisfeiler-Leman算法，是中央子程序的方法图同构问题。以迭代的方式，颜色细化计算其输入图的顶点的着色。n阶图色精化迭代次数的平凡上界为n-1。我们证明这个界是紧的。更准确地说，我们证明了通过明确的建设，有无穷多个图G上的颜色细化需要|G| -1迭代稳定。修改我们提出的无限家庭，我们表明，对于每个自然数n >= 10，有图的n个顶点上的颜色细化需要至少n-2次迭代，以达到稳定。
英文摘要：
The Colour Refinement procedure and its generalisation to higher dimensions, the Weisfeiler-Leman algorithm, are central subroutines in approaches to the graph isomorphism problem. In an iterative fashion, Colour Refinement computes a colouring of the vertices of its input graph. A trivial upper bound on the iteration number of Colour Refinement on graphs of order n is n-1. We show that this bound is tight. More precisely, we prove via explicit constructions that there are infinitely many graphs G on which Colour Refinement takes |G|-1 iterations to stabilise. Modifying the infinite families that we present, we show that for every natural number n >= 10, there are graphs on n vertices on which Colour Refinement requires at least n-2 iterations to reach stabilisation.
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。
