AI 中文总结
受纽结三色性启发引入超图精益着色概念及精益数，给出问题的两种替代形式，得到不同类型超图精益数界限，包含脚本实现算法，还给出未来方向及130个纽结和链环的精益数。
AI 中文摘要
受纽结三色性概念的启发，我们引入了超图的精益着色概念以及相关的超图精益数。精益着色通常使用很少的颜色，但仍需要常规图着色方法，这使得整体复杂度为NP - 难。我们提供了精益着色问题的两种替代形式，涉及抽象单纯复形上的一种着色和二分图上的部分着色。然后给出了\(k\) - 均匀、\(k\) - 部、宽路径连通或\(r\) - 完全超图的精益数界限。包含类似Python的脚本以实现和研究精益着色算法。最后给出了未来工作的方向并给出了130个纽结和链环的精益数。
英文摘要
Inspired by the notion of tricolorability of knots, we introduce the concept of lean coloring for hypergraphs and the associated lean number of a hypergraph. Lean coloring often involves very few colors, yet still requires the methods of usual graph coloring, forcing the overall complexity to be NP-Hard. We provide two alternative formulations of the lean coloring problem that involve a type of coloring on abstract simplicial complexes and a partial coloring on bipartite graphs. We then provide bounds for the lean numbers of hypergraphs that are $k$-uniform, $k$-partite, wide-path connected, or $r$-complete. Python-like script is included to allow the implementation and study of a lean coloring algorithm. We conclude with some directions for future work and present the lean numbers of $130$ knots and links.
Comments22 pages, 69 figures
