发表机构
The Graduate Center of the City University of New York; Amirkabir University of Technology; University of Tehran（纽约市立大学研究生中心; 阿米尔卡比尔理工大学; 德黑兰大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对超立方体上的配对防御银色染色问题，证明了最大颜色数约为普通银色染色的一半，并构造了达到界限的染色，刻画了极值结构。
AI 中文摘要
银色染色是一种正常染色，其中每个颜色都出现在一个指定的独立集的每个顶点的闭邻域中。当顶点被逐个测试时，这个局部条件就足够了。我们研究同时测试的更强要求：当独立集的一个或两个顶点同时受到攻击时，每种颜色必须为它们提供不同的邻近防御者。我们称此为配对防御银色染色。对于超立方体 $Q_d$，以一个奇偶类作为受攻击集，我们证明配对防御银色染色的最大颜色数至多为 $\u230a(d+3)/2\u230b$，大约是普通银色染色目标 $d+1$ 的一半。我们构造了在二的幂周围的四个连续维度上达到此界限的染色，并研究了达到界限的极值染色的结构。在每个奇数临界维度中，我们通过将防御奇偶类划分为半立方体的正则无三角形子图来刻画极值染色。这种刻画也有局部形式，涉及坐标匹配和缺陷坐标。
英文摘要
A silver colouring is a proper colouring in which every colour appears in the closed neighbourhood of each vertex of a prescribed independent set. This local condition suffices when vertices are tested one at a time. We study a stronger requirement for simultaneous testing: whenever one or two vertices of the independent set are attacked together, each colour must supply distinct nearby defenders for them. We call this a pair-defensive silver colouring. For the hypercube $Q_d$, with one parity class as the attacked set, we prove that the maximum number of colours in a pair-defensive silver colouring is at most $\lfloor(d+3)/2\rfloor$, roughly half the ordinary silver-colouring target $d+1$. We construct colourings attaining this bound in four consecutive dimensions around every power of two, and we study the structure of the extremal, bound-attaining colourings. In each odd critical dimension, we characterize the extremal colourings by a partition of the defender parity into regular, triangle-free subgraphs of the halved cube. This characterization also has a local form in terms of coordinate matchings and a defect coordinate
