On the Classical and Parameterized Complexity of Strong Odd Coloring

2026-10-01

AI Summary

This project systematically investigates the classical and parameterized complexity of strong odd coloring in graph theory. Methodologically, it integrates computational complexity theory, dynamic programming, and the Strong Exponential Time Hypothesis (SETH) to uncover the intrinsic hardness on perfectly orderable bipartite graphs and derive exact complexity bounds. The primary contributions include establishing NP-completeness and inapproximability for specific graph classes, proving the nonexistence of polynomial kernels alongside W[1]-hardness, and proposing a linear-time optimal algorithm for block graphs as well as a treewidth-based FPT algorithm. Overall, this work provides a complete characterization of the computational hardness of the problem, offering novel insights and an efficient solving framework for parameterized graph coloring.

Abstract

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.

Problem

• Strong odd coloring
• Parameterized complexity
• NP-completeness
• Treewidth
• Clique-width

Innovation

• strong odd coloring
• parameterized complexity
• tree-width
• approximation hardness
• FPT algorithm