The Colour Refinement algorithm is a classical procedure to detect symmetries in graphs, whose most prominent application is in graph-isomorphism tests. The algorithm and its generalisation, the Weisfeiler-Leman algorithm, evaluate local information to compute a colouring for the vertices in an iterative fashion. Different final colours of two vertices certify that no isomorphism can map one onto the other. The number of iterations that the algorithm takes to terminate is its central complexity parameter. For a long time, it was open whether graphs that take the maximum theoretically possible number of Colour Refinement iterations actually exist. Starting from an exhaustive search on graphs of low degrees, Kiefer and McKay proved the existence of infinite families of such long-refinement graphs with degrees 2 and 3, thereby showing that the trivial upper bound on the iteration number of Colour Refinement is tight. In this work, we provide a complete characterisation of the long-refinement graphs with low (or, equivalently, high) degrees. We show that, with one exception, the aforementioned families are the only long-refinement graphs with maximum degree at most 3, and we fully classify the long-refinement graphs with maximum degree 4. To this end, via a reverse-engineering approach, we show that all low-degree long-refinement graphs can be represented as compact strings, and we derive multiple structural insights from this surprising fact. Since long-refinement graphs are closed under taking edge complements, this also yields a classification of long-refinement graphs with high degrees. Kiefer and McKay initiated a search for long-refinement graphs that are only distinguished in the last iteration of Colour Refinement before termination. We conclude it in this submission by showing that such graphs cannot exist.

翻译：着色细化算法是一种用于检测图中对称性的经典过程，其最突出的应用在于图同构测试。该算法及其推广——Weisfeiler-Leman算法——通过评估局部信息以迭代方式计算顶点的着色。两个顶点最终着色的不同可以证明不存在同构能将其中一个映射到另一个。算法终止所需的迭代次数是其核心复杂度参数。长期以来，是否存在需要理论最大可能次数的着色细化迭代的图一直是一个悬而未决的问题。Kiefer和McKay从对低度图的穷举搜索出发，证明了度数为2和3的此类长细化图的无限族的存在性，从而表明着色细化迭代次数的平凡上界是紧的。在本工作中，我们完全刻画了具有低度（或等价地，高度）的长细化图。我们证明，除一个例外情况外，上述族是最大度数不超过3的唯一长细化图，并且我们完整分类了最大度数为4的长细化图。为此，通过一种逆向工程方法，我们表明所有低度长细化图都可以表示为紧凑的字符串，并从这个令人惊讶的事实中推导出多个结构性见解。由于长细化图在取边补集下是封闭的，这也产生了高度长细化图的分类。Kiefer和McKay曾发起寻找仅在着色细化终止前的最后一次迭代中才被区分的特殊长细化图。我们在本文中通过证明此类图不可能存在来终结这一搜索。