平衡汉明图与网格的精确多数C-着色
Exact majority C-colourings of balanced Hamming graphs and grids
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中文总结 AI 辅助
本文精确确定了平衡汉明图与网格的多数C-着色最大类数，给出下界构造与上界证明，并反驳了相关猜想。
中文摘要 AI 辅助
多数C-着色将图划分为若干类，使得每个顶点至少有半数邻居位于同一类中。记$M(G)$为最大类数。我们确定$M(K_q^{\square(2k+1)})=\left\lfloor\frac{q^{k+1}}{\lfloor q/2\rfloor+1}\right\rfloor\qquad(q\ge3,\\ k\ge0).$下界由显式矩形划分和均匀三维桥给出。一个穿孔矩形构造同时提供桥和中间维度的划分。上界是汉明边等周性的经典推论，文中包含一个初等证明。我们还证明$M(C_m\square P_n)=\frac n2\lfloor m/2\rfloor$对于$m\ge4$和偶数$n\ge2$成立，这反驳了arXiv:2608.27669v1中猜想4关于奇数$m\ge7$和偶数$n\ge6$的圆柱断言。进一步的层界确定了额外的圆柱和环面族。
英文摘要
A majority C-colouring partitions a graph into classes in which every vertex has at least half of its neighbours. Write $M(G)$ for the maximum number of classes. We determine $M(K_q^{\square(2k+1)})=\left\lfloor\frac{q^{k+1}}{\lfloor q/2\rfloor+1}\right\rfloor\qquad(q\ge3,\ k\ge0).$ The lower bound follows from explicit rectangular partitions and a uniform three-dimensional bridge. A punctured-rectangle construction supplies both the bridge and a partition in the intermediate dimension. The upper bound is a classical consequence of Hamming edge isoperimetry; an elementary proof is included. We also prove $M(C_m\square P_n)=\frac n2\lfloor m/2\rfloor$ for $m\ge4$ and even $n\ge2$, contradicting the cylinder assertion of Conjecture 4 in arXiv:2608.27669v1 for odd $m\ge7$ and even $n\ge6$. Further layer bounds determine additional cylinder and torus families.
