On the Parameterized Complexity of Odd Coloring
Sriram Bhyravarapu, Swati Kumari, I. Vinod Reddy
TL;DR
This work analyzes the parameterized complexity of Odd Coloring, where each vertex must have a neighborhood color that appears an odd number of times. The authors map a comprehensive landscape across multiple graph parameters, delivering a polynomial kernel for distance to clique, a kernel lower-bound for vertex cover, and fixed-parameter tractability results for distance to cluster, distance to co-cluster, and neighborhood diversity, while proving W[1]-hardness for clique-width. They also show polynomial-time solvability on certain restricted classes (e.g., cographs and split graphs) and NP-hardness on specific bipartite subclasses. Overall, the results delineate when Odd Coloring is tractable under structural constraints and highlight inherent hardness in key width- and cover-based parameters, guiding practical algorithm design for specialized graph families.
Abstract
A proper vertex coloring of a connected graph $G$ is called an odd coloring if, for every vertex $v$ in $G$, there exists a color that appears odd number of times in the open neighborhood of $v$. The minimum number of colors required to obtain an odd coloring of $G$ is called the \emph{odd chromatic number} of $G$, denoted by $χ_{o}(G)$. Determining $χ_o(G)$ known to be ${\sf NP}$-hard. Given a graph $G$ and an integer $k$, the \odc{} problem is to decide whether $χ_o(G)$ is at most $k$. In this paper, we study the parameterized complexity of the problem, particularly with respect to structural graph parameters. We obtain the following results: \begin{itemize} \item We prove that the problem admits a polynomial kernel when parameterized by the distance to clique. \item We show that the problem cannot have a polynomial kernel when parameterized by the vertex cover number unless ${\sf NP} \subseteq {\sf Co {\text -} NP/poly}$. \item We show that the problem is fixed-parameter tractable when parameterized by distance to cluster, distance to co-cluster, or neighborhood diversity. \item We show that the problem is ${\sf W[1]}$-hard parameterized by clique-width. \end{itemize} Finally, we study the complexity of the problem on restricted graph classes. We show that it can be solved in polynomial time on cographs and split graphs but remains NP-complete on certain subclasses of bipartite graphs.