从b-着色到$b^*$-着色：大围长与参数化复杂性
From b-Coloring to $b^*$-Coloring: Large Girth and Parameterized Complexity
- Faculty of Informatics, Masaryk University（马萨里克大学信息学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究从b-着色扩展到$b^*$-着色，证明围长≥7的图是$b^*$-单调的，发现围长≥5的d-正则图$b^*$-色数为d+1，还明确了$b^*$-着色的参数化复杂性与b-着色的关联及特定参数下算法的差异。
AI中文摘要：
b-着色是一种正常顶点着色，其中每个颜色类都包含一个被称为b-顶点的顶点，该顶点在其闭邻域中包含所有颜色。这类着色已从结构和算法角度得到广泛研究。最近，Zaker[DAM 2025]引入了$b^*$-着色的概念，它是一种b-着色，其中存在一个顶点，在其闭邻域中包含每种颜色的一个b-顶点。$b^*$-色数是存在使用k种颜色的$b^*$-着色时k的最大整数。我们部分回答了Zaker提出的问题，证明围长至少为7的图是$b^*$-单调的，即$b^*$-色数不会因取诱导子图而增大。此外，我们发现了一类围长至少为5的d-正则图，其$b^*$-色数为d+1，这强化了Dettlaff、Furmańczyk、Peterin、Roux和Ziemann[AMC 2024]关于b-着色的结果。我们还研究了寻找$b^*$-着色的参数化复杂性，表明对于许多结构参数，其复杂性与寻找b-着色的复杂性一致。特别是，在任何有界团宽类上，$b^*$-色数都可以在多项式时间内计算。对于大多数参数，从b-着色的转换是直接的，但对于反馈边数，$b^*$-着色的FPT算法实际上比Balabán[MFCS 2026]提出的b-着色算法简单得多。
英文摘要：
A b-coloring is a proper vertex coloring such that every color class contains a vertex, a so-called b-vertex, which sees all colors in its closed neighborhood. This type of coloring has been intensively studied from both structural and algorithmic point of view. Recently, Zaker [DAM 2025] introduced the notion of a b*-coloring, which is a b-coloring in which there is a vertex that sees a b-vertex of every color in its closed neighborhood. The b*-chromatic number is the maximum integer k such that there is a b*-coloring with k colors. We partially answer a question posed by Zaker and prove that graphs of girth at least 7 are b*-monotonic, which means that the b*-chromatic number does not increase by taking an induced subgraph. In addition, we discover a class of d-regular graphs of girth at least 5 with b*-chromatic number d+1, which strengthens a result about b-colorings by Dettlaff, Furmańczyk, Peterin, Roux, and Ziemann [AMC 2024]. We also study the parameterized complexity of finding b*-colorings, and show that for many structural parameters, the complexity coincides with that of finding b-colorings. In particular, the b*-chromatic number can be computed in polynomial time on any class of bounded clique-width. For most parameters, the translation from b-colorings is straightforward but for the feedback edge number, the FPT algorithm for b*-colorings is actually much simpler than that for b-colorings by Balabán [MFCS 2026].
