On the Parameterized Complexity of Odd Coloring

Abstract

A proper vertex coloring of a connected graph G is called an odd coloring if, for every vertex v of G there is 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 odd chromatic number of G and it is denoted by $\chi _{o}(G)$. The problem of determining $\chi _o(G)$ is $\textsf{NP}$-hard. Given a graph G and an integer k, the Odd Coloring problem is to decide whether $\chi _o(G)$ is at most k. In this paper, we study the problem from the viewpoint of parameterized complexity. In particular, we study the problem with respect to structural graph parameters. We prove that the problem admits a polynomial kernel when parameterized by distance to clique. On the other hand, we show that the problem cannot have a polynomial kernel when parameterized by vertex cover number unless $\textsf{NP} \subseteq \mathsf{Co {\text{- }} NP/poly}$. We show that the problem is fixed-parameter tractable when parameterized by distance to cluster, distance to co-cluster, or neighborhood diversity. We show that the problem is $\mathsf{W[1]}$-hard parameterized by clique-width. Finally, we study the problem on restricted graph classes. We show that the problem can be solved in polynomial time on cographs and split graphs.