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

Abstract: 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 $$\max{1,2^{k-2}^w\cdot} n<sup>{k+O(1)}$$. 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.

• The paper presents an XP dynamic-programming algorithm for equitable k-colorings on graphs of bounded clique-width.
• The paper is capped with a refined algorithm for bounded linear clique-width and a subexponential counting model for 3-coloring $P_t-free graphs.
• A SETH conditional lower bound demonstrates the boundary of exponential parameters for clique-width in this specific context.

This paper studies the algorithmic complexity of equitable $k$-colorings—proper $k$-colorings whose color classes differ in size by at most one—under structural parameterizations. Its main contributions are an $\mathsf{XP}$ dynamic-programming algorithm for counting equitable $k$-colorings on graphs of bounded clique-width, a matching-in-spirit $\mathsf{SETH}$ lower bound showing that the exponential dependence on clique-width is essentially tight, a refined algorithm for bounded linear clique-width with base $(2^k-2)$ instead of a $2^{O(k)}$ factor, and a subexponential-time counting algorithm for equitable list 3-colorings of $P_t$-free graphs (2606.27159).

Background and problem statement

A proper $k$-coloring $\varphi$ of a graph $k$0 is equitable if $k$1 for all colors $k$2, where $k$3. The decision problem Equitable $k$4-Coloring has a mixed complexity landscape: it is polynomial-time solvable on graphs of bounded treewidth (Bodlaender and Fomin), yet $k$5-hard already on cographs, hence on graph classes of constant clique-width; Fellows et al. further showed $k$6-hardness parameterized by treewidth plus the number of colors. This creates an apparent tension: for variable $k$7 the problem is hard even at constant clique-width, while the paper shows that freezing $k$8 restores tractability, even for the counting version.

Counting on bounded clique-width

The central positive result is as follows: given an integer $k$9 and an $\mathsf{XP}$0-vertex graph $\mathsf{XP}$1 together with a $\mathsf{XP}$2-expression constructing $\mathsf{XP}$3, the number of equitable $\mathsf{XP}$4-colorings can be computed in time

$\mathsf{XP}$5

so for every fixed $\mathsf{XP}$6 the problem lies in $\mathsf{XP}$7 on classes of bounded clique-width, provided a clique-width expression is supplied. The algorithm performs bottom-up dynamic programming over the syntax tree of the expression using a signature: a pair recording, for each label $\mathsf{XP}$8, the set $\mathsf{XP}$9 of colors appearing on that label class, and for each color $k$0, the global size $k$1 of its color class. The global size vector is essential—ordinary coloring DP needs only local information about label–color interactions, whereas equitability constrains class sizes globally and must be enforced at the root. The root condition follows from uniqueness of Euclidean division: writing $k$2, a proper coloring is equitable exactly when each $k$3 and precisely $k$4 colors have class size $k$5. Union nodes require convolution over child signatures ($k$6 per node where $k$7), join nodes check disjointness of label color sets, and relabel nodes admit at most $k$8 preimages, giving the stated bound up to polynomial factors for arithmetic on integers of $k$9 bits.

An immediate consequence is that the $\mathsf{SETH}$0-hardness of the problem for combined parameter clique-width plus $\mathsf{SETH}$1 is driven entirely by variability of $\mathsf{SETH}$2: fixing $\mathsf{SETH}$3 places both decision and counting in polynomial time on bounded-clique-width classes. Note the dependence on being given the expression; computing an optimal clique-width decomposition is not addressed.

A SETH lower bound

The upper bound's exponential dependence on width is complemented by a conditional lower bound: for every fixed $\mathsf{SETH}$4 and every $\mathsf{SETH}$5, unless $\mathsf{SETH}$6 fails, Equitable $\mathsf{SETH}$7-Coloring cannot be solved in time $\mathsf{SETH}$8. The reduction from Lampis's SETH lower bound for $\mathsf{SETH}$9-Coloring on clique-width pads the input graph $(2^k-2)$0 with an independent set of size $(2^k-2)$1. Any proper $(2^k-2)$2-coloring of $(2^k-2)$3 extends to one of $(2^k-2)$4 in which every color class has size exactly $(2^k-2)$5 (each color receives $(2^k-2)$6 padding vertices), so proper $(2^k-2)$7-colorability of $(2^k-2)$8 coincides with equitably $(2^k-2)$9-colorability of $2^{O(k)}$0; conversely any equitable coloring of $2^{O(k)}$1 restricts to a proper coloring of $2^{O(k)}$2. Adding the independent vertices via a fresh label increases clique-width by at most one, transferring the lower bound. Two caveats apply: the lower bound does not fully match the $2^{O(k)}$3 upper bound—it rules out improving only the base-of-exponentiation in $2^{O(k)}$4 for fixed $2^{O(k)}$5—and it concerns the decision problem rather than counting.

Refinement for linear clique-width

For graphs given by a linear $2^{O(k)}$6-expression, processing the expression left-to-right yields a sharper running time:

$2^{O(k)}$7

The improvement comes from tracking only live labels: a label is live at time $2^{O(k)}$8 if its class is nonempty and some future join operation will add edges incident to its descendants, meaning the colors currently used on it still matter for properness. Non-live labels are recorded as $2^{O(k)}$9 and ignored. The key structural lemma is that any extendible coloring uses a strict nonempty subset of colors on every live label class—a live label whose class uses all $P_t$0 colors cannot survive the witnessing join—so states restrict to $P_t$1 possibilities rather than $P_t$2, and colorings violating this condition can be safely discarded. Correct handling of relabel operations requires several claims about how liveness evolves under add, join, and relabel steps (e.g., liveness is monotone under joins, and a newly live label under relabel forces the source class to be empty or the source label to have been live). Liveness for all timestamps is computable in $P_t$3. For fixed $P_t$4, this gives polynomial time on classes of bounded linear clique-width, with a strictly better base than the general clique-width algorithm.

Subexponential counting on $P_t$5-free graphs

The final result moves beyond clique-width: for every fixed $P_t$6, the number of equitable list 3-colorings of a $P_t$7-free graph on $P_t$8 vertices can be computed in time $P_t$9. This setting is incomparable to bounded clique-width since $k$0-free graphs already have unbounded clique-width. The proof reduces to partition-function computation: the coefficient extraction identity shows that the number of list 3-colorings with prescribed color-class sizes $k$1 is the coefficient of $k$2 in the partition polynomial $k$3, and equitable colorings correspond to the at most three coefficients with sizes $k$4 or $k$5. Since $k$6 satisfies the neighborhood condition of Groenland et al.'s subexponential algorithm for $k$7-coloring $k$8-free graphs, that theorem supplies the polynomial; extracting $k$9 monomials with $\varphi$0-bit coefficients is then polynomial. Trivial cases ($\varphi$1, where the graph is edgeless or a union of cliques) are handled directly. The restriction to three colors is inherited from the underlying $\varphi$2-coloring machinery, and the result covers the list variant, which is stronger than the uncolored case.

Limitations and open questions

Several qualifications are explicit in the paper. All clique-width algorithms assume the input includes a (near-)optimal expression; the complexity of finding such decompositions is not treated, nor is whether the linear-clique-width bound extends in full strength when only a general expression is available. The SETH lower bound matches neither the full $\varphi$3 term of the upper bound nor applies to counting, leaving open whether the exponent on $\varphi$4 or the base $\varphi$5 is tight in stronger senses. On the structural side, the subexponential result is confined to $\varphi$6; extending it to larger numbers of colors would require machinery beyond the Groenland et al. framework. Finally, the paper notes that the Chen–Lih–Wu conjecture on equitable $\varphi$7-coloring remains open in general, providing context but no resolution.

Conclusion

The paper resolves the parameterized complexity of Equitable $\varphi$8-Coloring along the combined parameter $\varphi$9 plus clique-width in a fairly complete sense: $k$00-hard for variable $k$01 (by prior work), yet in $k$02—indeed polynomial for fixed $k$03—for both decision and counting once $k$04 is frozen, with a SETH-based lower bound showing the exponential dependence on clique-width is near-optimal. The linear-clique-width refinement and the subexponential algorithm for $k$05-free graphs demonstrate that the equitable constraint, though globally coupling color-class sizes, remains amenable to the same structural techniques that succeed for ordinary coloring.