发表机构
Zhejiang University of Science and Technology; Hangzhou Normal University; Beijing University of Technology（浙江科技学院; 杭州师范大学; 北京工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究推导了B-着色的退化界、稳定性及严格间隙，明确了$d$-退化图和无环多重图的$q_B(G)$上界，确定了无环多重图$q_B(G)$的数值间隙。
AI 中文摘要
图的B-着色是一种恰当边着色，其中每个4-环都是彩虹的，$q_B(G)$表示这种着色中所需的最少颜色数。设$\triangle_2(G)$为图$G$中两个不同顶点的公共邻居的最大数量。我们证明，对于满足$1\leq d \triangle$的整数$d$，每个最大度为$\triangle(G)\triangle$的有限简单$d$-退化图$G$都满足$q_B(G)\triangle+(d-1)\triangle_2(G)\triangle d\triangle$。因此，$d\triangle$是精确最大值，当且仅当图包含$K_{d,\triangle}$时取等号。更一般地，若$q_B(G)\triangle d\triangle-s$（其中$0\triangle$），则$G$包含$K_{d,\triangle-s}$；若同时$s\triangle$，则$G$至少有$d-s$个具有相同开放邻域的最大度顶点。对于$\triangle\uc0b3$，我们进一步证明，每个不含$K_{3,\triangle}$的3-退化图满足$q_B(G)\uc0b3\triangle-2$；而$K_{3,\triangle-1}$的例子表明该界的误差不超过1。对于无环多重图，我们确定了$q_B(G)$可能值中的严格间隙：对于每个整数$\triangle\uc0b3$，每个最大度为$\triangle(G)\triangle$的有限无环多重图$G$，若其没有同构于$K_{\triangle,\triangle}$的连通分支，则满足$q_B(G)\uc0b3\triangle(\triangle-1)$；若存在这样的连通分支，则$q_B(G)\uc0b3\triangle^2$。$K_{\triangle,\triangle-1}$和$K_{\triangle,\triangle}-e$都能达到界$\triangle(\triangle-1)$。因此，在最大度不超过$\triangle$的有限无环多重图中，$q_B(G)$不存在严格介于$\triangle^2-\triangle$和$\triangle^2$之间的值。
英文摘要
A $B$-coloring of a graph is a proper edge-coloring in which every $4$-cycle is rainbow, and $q_B(G)$ denotes the minimum number of colors in such a coloring. Let $Δ_2(G)$ denote the maximum number of common neighbors of two distinct vertices of $G$. We prove that, for integers $1\le d\leΔ$, every finite simple $d$-degenerate graph $G$ with $Δ(G)\leΔ$ satisfies $$q_B(G)\le Δ+(d-1)Δ_2(G)\le dΔ.$$ Consequently, $dΔ$ is the exact maximum, with equality precisely for graphs containing $K_{d,Δ}$. More generally, if $q_B(G)\ge dΔ-s$, where $0\le s<Δ$, then $G$ contains $K_{d,Δ-s}$; if also $s<d$, then $G$ has at least $d-s$ vertices of degree $Δ$ with the same open neighborhood. For $Δ\ge3$, we further show that every $K_{3,Δ}$-free 3-degenerate graph satisfies $q_B(G)\le3Δ-2$; the example $K_{3,Δ-1}$ shows that this bound is best possible up to one. For loopless multigraphs, we establish a sharp gap in the possible values of $q_B(G)$. For every integer $Δ\ge3$, every finite loopless multigraph $G$ with $Δ(G)\leΔ$ satisfies $$q_B(G)\leΔ(Δ-1)$$ unless $G$ has a component isomorphic to $K_{Δ,Δ}$, in which case $q_B(G)=Δ^2$. The bound $Δ(Δ-1)$ is attained by both $K_{Δ,Δ-1}$ and $K_{Δ,Δ}-e$. Consequently, among finite loopless multigraphs with maximum degree at most $Δ$, no value of $q_B(G)$ lies strictly between $Δ^2-Δ$ and $Δ^2$.
