发表机构
Indian Statistical Institute（印度统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究将强边着色的6-近似算法从单位圆盘图推广到更一般的圆盘图，并将单位圆盘图的强色指数上界改进至225/142，同时证明该界对强列表色指数也成立。
AI 中文摘要
图$G$的强边着色是一种边着色，其中每个颜色类都是一个诱导匹配。所需的最少颜色数称为强色指数$\chi'_s(G)$。如果每条边$e$被分配一个列表$L'(e)$，并且其颜色必须属于$L'(e)$，则相应的参数称为强列表色指数$\chi'_{s,\ell}(G)$。根据定义，$\chi'_s(G)\le\chi'_{s,\ell}(G)$。Barrett等人给出了单位圆盘图上强边着色的8-近似算法，Grelier等人将近似因子改进到6。我们的第一个结果将这一因子为6的保证从单位圆盘图扩展到严格更大的圆盘图类。在另一个方向上，Erdős和Nešetřil猜想最大度为$\Delta$的图的强色指数渐近地至多为$1.25\Delta^2$。已发表的最佳一般渐近上界的前导系数为$1.772$，由Hurley等人给出。对于单位圆盘图，Dębski等人证明了$\chi'_s(G) \leq 1.625 \Delta^2$。我们的第二个结果将这一前导系数改进到$225/142 \approx 1.5845$。事实上，该证明建立了单位圆盘图更强的界$\chi'_{s,\ell}(G)\le\frac{225}{142} \Delta^2+O(\Delta)$。
英文摘要
A strong edge colouring of a graph $G$ is an edge colouring in which every colour class is an induced matching. The minimum number of colours is the strong chromatic index $χ'_s(G)$. If each edge $e$ is assigned a list $L'(e)$ and its colour must belong to $L'(e)$, the corresponding parameter is the strong list chromatic index $χ'_{s,\ell}(G)$. From the definitions, $χ'_s(G)\leχ'_{s,\ell}(G)$. Barrett et al. gave an $8$-approximation for strong edge colouring on unit disk graphs and Grelier et al. improved the approximation factor to $6$. Our first result extends this factor-$6$ guarantee from unit disk graphs to the strictly larger class of disk graphs. In another direction, Erdős and Nešetřil conjectured that the strong chromatic index of a graph of maximum degree $Δ$ is asymptotically at most $1.25Δ^2$. The best published general asymptotic upper bound has leading coefficient $1.772$, due to Hurley et al. For unit disk graphs, Dębski et al. proved that $χ'_s(G) \leq 1.625 Δ^2$. Our second result improves this leading coefficient to $225/142 \approx 1.5845$. In fact, the proof establishes a stronger bound $χ'_{s,\ell}(G)\le\frac{225}{142} Δ^2+O(Δ)$ for unit disk graphs.
Comments14 pages, 2 figures
