Counting equitable $k$-colorings in graphs of bounded clique-width

For a graph $G$, a proper $k$-coloring of $G$ is \emph{equitable} if the sizes of any two color classes differ by at most one. The \textsc{Equitable $k$-Coloring} problem asks, for a given graph $G$ and integer $k$, whether $G$ admits an equitable $k$-coloring. Bodlaender and Fomin showed that it is polynomial-time solvable on graphs of bounded treewidth, while it remains $\NP$-hard on cographs, and thus on graphs of constant clique-width. Fellows et al. showed that the problem becomes $\mathsf{W[1]}$-hard when parameterized by tree-width (and hence clique-width) plus the number of colors~$k$. We first show that, for every fixed $k$, counting equitable $k$-colorings is polynomial-time solvable on graph classes of bounded clique-width, given a clique-width expression. We then show that, under $\mathsf{SETH}$, the dependence on clique-width in this algorithm is essentially optimal. As a consequence, our results provide a fairly tight picture of the complexity of \textsc{Equitable $k$-Coloring} with respect to the combined parameter $k$+clique-width. Second, we refine our clique-width algorithm for the linear setting. We show that there exists an algorithm, given an integer $k\ge 1$ and an $n$-vertex graph $G$ together with a linear $w$-expression constructing $G$, computes the number of equitable $k$-colorings of $G$ in time $\max\{1,2^k-2\}^w\cdot n^{k+O(1)}$. Third, we consider a different structural restriction, namely the class of $P_t$-free graphs. A graph is called $P_t$-free if it does not contain the path on $t$ vertices as an induced subgraph. This is a different setting from bounded clique-width; in particular, already $P_5$-free graphs have unbounded clique-width. Nevertheless, we show that for every $P_t$-free graph $G$, the number of equitable list $3$-colorings of $G$ can be computed in subexponential time.