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 $\chi_{o}(G)$. Determining $\chi_o(G)$ known to be ${\sf NP}$-hard. Given a graph $G$ and an integer $k$, the \odc{} problem is to decide whether $\chi_o(G)$ 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.

翻译：连通图$G$的一个正常顶点着色若满足：对于$G$中的每个顶点$v$，都存在一种颜色在$v$的开邻域中出现奇数次，则该着色称为奇着色。使$G$获得奇着色所需的最少颜色数称为$G$的\emph{奇色数}，记作$\chi_{o}(G)$。已知确定$\chi_o(G)$是${\sf NP}$-困难的。给定图$G$与整数$k$，\odc{}问题旨在判定$\chi_o(G)$是否不超过$k$。本文研究该问题的参数化复杂度，尤其关注结构图参数。我们获得以下结果：\begin{itemize} \item 证明当以距离团数（distance to clique）为参数时，该问题存在多项式核。\item 证明当以顶点覆盖数为参数时，除非${\sf NP} \subseteq {\sf Co {\text -} NP/poly}$，否则该问题不存在多项式核。\item 证明当以距离聚类数（distance to cluster）、距离余聚类数（distance to co-cluster）或邻域多样性（neighborhood diversity）为参数时，该问题是固定参数可解的。\item 证明当以团宽度（clique-width）为参数时，该问题是${\sf W[1]}$-困难的。\end{itemize} 最后，我们研究了该问题在受限图类上的复杂度。证明其在补图（cographs）和分裂图（split graphs）上可在多项式时间内求解，但在二分图的某些子类上仍保持NP完全性。