强奇染色的经典与参数化复杂度

中文摘要 AI 辅助

图$G$的强奇$k$-染色是一种正常$k$-染色，使得每个非孤立顶点的邻域中出现的每种颜色都出现奇数次。使得$G$允许强奇$k$-染色的最小$k$称为$G$的强奇色数，记为$\chi_{\text{so}}(G)$。给定图$G$和整数$k$，强奇$k$-可染色性问题询问$G$是否允许强奇$k$-染色。已知强奇$k$-可染色性在一般图中是NP完全的。在本文中，我们证明该问题在完美消去二分图（二分图的一个子类）上对于$k\geq3$是NP完全的。此外，我们证明对于每个$\varepsilon>0$，$\chi_{\text{so}}(G)$无法在因子$O(n^{\frac{1}{2}-\varepsilon})$内近似。在积极方面，我们获得了一个线性时间算法来计算块图的最优强奇染色。从参数化角度来看，我们提出了一个以树宽为参数的强奇$k$-可染色性的FPT算法。此外，我们证明在SETH假设下，以树宽为参数时，该问题无法在时间$(k-\varepsilon)^{\texttt{tw}}n^{O(1)}$内解决，其中$k\geq3$且$\varepsilon>0$。进一步，我们证明强奇$k$-可染色性在以反馈顶点集为参数时不允许多项式核。最后，我们证明强奇$k$-可染色性在以团宽为参数时是W[1]-难的。

英文摘要

A strong odd $k$-coloring of a graph $G$ is a proper $k$-coloring such that every color appearing in the neighborhood of a non-isolated vertex appears an odd number of times. The minimum $k$ for which $G$ admits a strong odd $k$-coloring is the \emph{strong odd chromatic number}, denoted by $χ_{\text{so}}(G)$, of $G$. Given a graph $G$ and an integer $k$, \textsc{strong odd $k$-colorability} problem asks whether $G$ admits a strong odd $k$-coloring. It is known that STRONG ODD $k$-COLORABILITY is NP-complete in general graphs. In this paper, we prove that the problem is NP-complete on perfect elimination bipartite graphs for $k\geq3$, which is a subclass of bipartite graphs. Furthermore, we show that $χ_{\text{so}}(G)$ is inapproximable within a factor of $O(n^{\frac{1}{2}-\varepsilon})$ for every $\varepsilon>0$. On the positive side, we obtain a linear time algorithm to compute an optimal strong odd coloring for block graphs. From a parameterized perspective, we present an FPT algorithm for STRONG ODD $k$-COLORABILITY when parameterized by treewidth. Moreover, we show that the problem cannot be solved in time $(k-\varepsilon)^{\texttt{tw}}n^{O(1)}$ for every $k\geq3$ and $\varepsilon>0$ when parameterized by treewidth under SETH. Furthermore, we show that STRONG ODD $k$-COLORABILITY does not admit a polynomial kernel when parameterized by feedback vertex set. Lastly, we prove that STRONG ODD $k$-COLORABILITY is W[1]-hard when parameterized by clique-width.

发表机构

• Indian Institute of Technology (ISM), Dhanbad（印度理工学院（ISM）丹巴德分校）
• Indian Institute of Technology, Jodhpur, India（印度理工学院乔德普尔分校）

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