Structural Parameterizations for Equitable Coloring

An $n$-vertex graph is equitably $k$-colorable if there is a proper coloring of its vertices such that each color is used either $\left\lfloor n/k \right\rfloor$ or $\left\lceil n/k \right\rceil$ times. While classic Vertex Coloring is fixed parameter tractable under well established parameters such as pathwidth and feedback vertex set, Equitable Coloring is $\mathsf{W}[1]$-$\mathsf{hard}$. We present an extensive study of structural parameterizations of Equitable Coloring, tackling both tractability and kernelization questions. We begin by showing that the problem is fixed parameter tractable when parameterized by distance to cluster or by distance to co-cluster -- improving on the $\mathsf{FPT}$ algorithm of Fiala et al. [Theoretical Computer Science, 2011] parameterized by vertex cover -- and also when parameterized by distance to disjoint paths of bounded length. To justify the latter result, we adapt a proof of Fellows et al. [Information and Computation, 2011] to show that Equitable Coloring is $\mathsf{W}[1]$-$\mathsf{hard}$ when simultaneously parameterized by distance to disjoint paths and number of colors. In terms of kernelization, on the positive side we present a linear kernel for the distance to clique parameter and a cubic kernel when parameterized by the maximum leaf number; on the other hand, we show that, unlike Vertex Coloring, Equitable Coloring does not admit a polynomial kernel when jointly parameterized by vertex cover and number of colors, unless $\mathsf{NP} \subseteq \mathsf{coNP}/\mathsf{poly}$. We also revisit the literature and derive other results on the parameterized complexity of the problem through minor reductions or other observations.