Majority C-coloring in Cartesian products
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 $\overline{\chi}_{\geqslant}(G)$. In this paper, the focus is on the majority C-coloring in Cartesian product graphs. It is shown that $\overline{\chi}_{\geqslant}(G \square H) \ge \overline{\chi}_{\geqslant}(G) \overline{\chi}_{\geqslant}(H)$ gives a sharp lower bound, but the difference also can be arbitrarily large. For two-dimensional Hamming graphs, the exact value $\overline{\chi}_{\geqslant}(K_m \square K_n) = \min\{m,n\}$ 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 $\overline{\chi}_{\geqslant}(K_n^{\square, k})= n^{k/2}$ holds for every even integer $k$. If $k$ is odd and the Hamming graph is the $k$-dimensional hypercube, then $\overline{\chi}_{\geqslant}(K_2^{\square, k})= 2^{\lfloor k/2\rfloor}$. 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 $\overline{\chi}_{\geqslant}(P_m \square P_n) = 1 + \lfloor m/2\rfloor \lfloor n/2\rfloor$ if at least one of $m$ and $n$ is odd, while $\overline{\chi}_{\geqslant}(P_m \square P_n)=mn/4$ holds if both parameters are even and $m \ge n \ge 4$. The paper concludes with a conjecture and several open problems.