AI 中文总结
本文确定了矩形、圆柱形、环形等网格图的B-着色最少颜色数，高维情形下也得到了相关精确结果与界。
AI 中文摘要
图G的B-着色是一种恰当边着色，其中每个4-圈都是彩虹的。令q_B(G)为此类着色所需的最少颜色数。Gyárfás与Sárközy（2023年）确定了当G为矩形网格P_m□P_n时的q_B(G)。本文中，我们完全确定了圆柱形和环形网格图G的q_B(G)。对于环形网格G=C_m□C_n，其中m、n≥3为整数，我们证明：若m和n均为偶数且G≠C_4□C_{4k+2}（k≥1为整数），则q_B(G)=4=Δ(G)；若m、n中至少有一个为奇数或G≅C_4□C_{4k+2}（k≥1为整数），则q_B(G)=5=Δ(G)+1。对于圆柱形网格G=C_s□P_m，q_B(G)同样取决于s的奇偶性与P_m的长度：当m≥2、n≥2时，q_B(C_{2n}□P_m)=4；当m≥2、n≥1时，若2≤m≤n，则q_B(C_{2n+1}□P_m)=4，若m≥n+1，则q_B(C_{2n+1}□P_m)=5。在高维情形下，我们讨论了离散环形与ℓ-圆柱形网格的B-着色，得到了若干精确结果与确定的界。
英文摘要
A B-coloring of a graph $G$ is a proper edge-coloring in which every $4$-cycle is rainbow. Let $q_B(G)$ be the minimum number of colors in such a coloring. Gyárfás and Sárközy (2023) determine $q_B(G)$ when $G=P_m\square P_n$ is a rectangular grid. In this paper, we completely determine $q_B(G)$ for cylindrical and torus grid graphs $G$. For a torus grid $G=C_m\square C_n$, where $m,n\ge3$ are integers, we prove that $q_B(G)=4=Δ(G)$ if both $m$ and $n$ are even and $G\not\cong C_4\square C_{4k+2}$ for any integer $k\ge1$, and that $q_B(G)=5=Δ(G)+1$ if at least one of $m,n$ is odd or $G\cong C_4\square C_{4k+2}$ for some integer $k\ge1$. For a cylindrical grid $G=C_s\square P_m$, $q_B(G)$ also depends on the parity of $s$ and the length of $P_m$. For integers $m\ge2$ and $n\ge2$, we have $q_B(C_{2n}\square P_m)=4$. For integers $m\ge2$ and $n\ge1$, we have $q_B(C_{2n+1}\square P_m)=4$ if $2\le m\le n$, whereas $q_B(C_{2n+1}\square P_m)=5$ if $m\ge n+1$. In higher dimensions, we discuss the B-coloring of discrete torus and $\ell$-cylindrical grid, obtaining some exact results and certain bounds.
Comments18 pages, 4 figures, 9 tables
