AI 中文总结
研究$K_{2,t}$-无平面图的B-染色数，证明在特定条件下$q_B(G)=\Delta(G)$，并给出一般上界，推广了已知的平面图结果。
AI 中文摘要
图$G$的B-染色是一种正常的边染色，使得每个$4$-圈都接收到四种不同的颜色；令$q_B(G)$为这种染色中所用颜色的最小数目。最大度为$\Delta$的每个图都是$K_{2,\Delta+1}$-无的；因此，已知的对于$\Delta\ge38$的平面图的$2\Delta$界（Kong等人，2026年）促使我们研究$K_{2,t}$-无平面图，其中$t\ge2$是整数。我们证明当$t=2$且$\Delta(G)\ge7$，或当$t\ge3$且$\Delta(G)\ge14(t-1)$时，$q_B(G)=\Delta(G)$。对于$t\ge35$，无论$\Delta(G)$如何，界$q_B(G)\le\Delta(G)+t-1$成立；对于每个$t\ge2$，当$\Delta(G)>428$时它也成立。最后，对于每个整数$k\ge1$，每个$k$-退化的$K_{2,t}$-无图满足$q_B(G)\le\Delta(G)+(k-1)\min\{t-1,\Delta(G)\}$，当$k\ge2$且$t-1\ge k$时，对于$K_{k,t-1}$等式成立。
英文摘要
A B-coloring of a graph $G$ is a proper edge-coloring in which every $4$-cycle receives four distinct colors; let $q_B(G)$ be the minimum number of colors in such a coloring. Every graph of maximum degree $Δ$ is $K_{2,Δ+1}$-free; hence the known $2Δ$ bound for planar graphs with $Δ\ge38$ (Kong et al., 2026) motivates our study of $K_{2,t}$-free planar graphs, where $t\ge2$ is an integer. We prove $q_B(G)=Δ(G)$ when $t=2$ and $Δ(G)\ge7$, or when $t\ge3$ and $Δ(G)\ge14(t-1)$. For $t\ge35$, the bound $q_B(G)\leΔ(G)+t-1$ holds regardless of $Δ(G)$; for every $t\ge2$, it also holds when $Δ(G)>428$. Finally, for every integer $k\ge1$, every $k$-degenerate $K_{2,t}$-free graph satisfies $q_B(G)\leΔ(G)+(k-1)\min\{t-1,Δ(G)\}$, with equality for $K_{k,t-1}$ when $k\ge2$ and $t-1\ge k$.
Comments13 pages, 6 figures
