AI 中文总结
本文研究笛卡尔积图的多数C-着色，给出其紧下界，确定二维汉明图等的精确值，提出笛卡尔网格的结果，文末附猜想与开放问题。
AI 中文摘要
图$G$的多数C-着色是指为其顶点分配颜色，使得每个顶点至少与一半的邻居拥有相同颜色，这种着色中$G$可使用的最大颜色数记为$\u0305\u03C7_{\u2265}(G)$。本文聚焦于笛卡尔积图的多数C-着色，证明$\u0305\u03C7_{\u2265}(G \u25A1 H) \u2265 \u0305\u03C7_{\u2265}(G) \u0305\u03C7_{\u2265}(H)$是一个紧下界，但二者差值可任意大；对二维汉明图，确定精确值$\u0305\u03C7_{\u2265}(K_m \u25A1 K_n) = \u007B m,n \u007D$；研究高维平衡汉明图（即完全图关于笛卡尔积的$k$次幂），证明对所有偶数$k$有$\u0305\u03C7_{\u2265}(K_n^{\u25A1, k})= n^{k/2}$，若$k$为奇数且汉明图是$k$维超立方体，则$\u0305\u03C7_{\u2265}(K_2^{\u25A1, k})= 2^{\u230A k/2\u230B}$；还对所有$n \u2265 7$和奇数$k \u2265 3$，给出$K_n^{\u25A1, k}$的多数C-着色方案，其颜色数至少为$3 n^{\u230A k/2\u230B}/2$；对笛卡尔网格，得到主要结果：若$m$和$n$中至少一个为奇数，则$\u0305\u03C7_{\u2265}(P_m \u25A1 P_n) = 1 + \u230A m/2\u230B \u230A n/2\u230B$，若二者均为偶数且$m \u2265 n \u2265 4$，则$\u0305\u03C7_{\u2265}(P_m \u25A1 P_n)=mn/4$；文末提出一个猜想和若干开放问题。
英文摘要
A majority C-coloring of a graph $G$ assigns colors to the vertices such that every vertex shares its color with at least half of its neighbors. The maximum number of colors that can be used in such a coloring of $G$ is denoted by $\overlineχ_{\geqslant}(G)$. In this paper, the focus is on the majority C-coloring in Cartesian product graphs. It is shown that $\overlineχ_{\geqslant}(G \square H) \ge \overlineχ_{\geqslant}(G) \overlineχ_{\geqslant}(H)$ gives a sharp lower bound, but the difference also can be arbitrarily large. For two-dimensional Hamming graphs, the exact value $\overlineχ_{\geqslant}(K_m \square K_n) = \min\{m,n\}$ is established. Balanced Hamming graphs of higher dimension, that is the $k$th powers of complete graphs with respect to the Cartesian product, are also studied. It is proved that $\overlineχ_{\geqslant}(K_n^{\square, k})= n^{k/2}$ holds for every even integer $k$. If $k$ is odd and the Hamming graph is the $k$-dimensional hypercube, then $\overlineχ_{\geqslant}(K_2^{\square, k})= 2^{\lfloor k/2\rfloor}$. On the other hand, a majority C-coloring of $K_n^{\square, k}$ with at least $3 n^{\lfloor k/2\rfloor}/2 $ colors is presented for every $n \ge 7$ and odd $k \ge 3$. For Cartesian grids, the main result shows that $\overlineχ_{\geqslant}(P_m \square P_n) = 1 + \lfloor m/2\rfloor \lfloor n/2\rfloor$ if at least one of $m$ and $n$ is odd, while $\overlineχ_{\geqslant}(P_m \square P_n)=mn/4$ holds if both parameters are even and $m \ge n \ge 4$. The paper concludes with a conjecture and several open problems.
