Kromatic Symmetric Function (KSF)

• Kromatic symmetric function is a K-theoretic analogue of the chromatic symmetric function that sums over proper set-valued colorings, with its lowest-degree component recovering the classical CSF.
• It features a rich algebraic structure including multiplicative properties, Hopf-algebraic and quasisymmetric formulations, enabling decompositions into augmented monomials, power-sum bases, and multifundamental functions.
• Recent studies reveal its nuanced positivity properties, distinguishing power beyond CSF for many graph pairs while also exhibiting limits, such as counterexamples and non-lifts of classical e-positivity conjectures.

The Kromatic symmetric function is a $K$-theoretic analogue of Stanley’s chromatic symmetric function. In the recent literature, it is typically denoted $\overline{X}_G$ (or $\overline{X}_{(G,\omega)}$ in the weighted case) and is defined by summing over proper set-valued colorings: each vertex receives a nonempty finite set of colors, and adjacent vertices receive disjoint color sets. It is symmetric but not homogeneous, and its lowest-degree homogeneous component is the classical chromatic symmetric function $X_G$. Earlier use of “KSF” was nonstandard, but the current meaning is this $K$-theoretic invariant introduced by Crew, Pechenik, and Spirkl (Crew et al., 2023), with later work placing it in a broader Hopf-algebraic and quasisymmetric framework (Dahlberg et al., 2019).

1. Definition and basic formalism

For a vertex-weighted graph $(G,\omega)$ with $\omega:V(G)\to \mathbb{N}$, a proper set coloring is a map

$$\kappa:V(G)\to \{S\subset \mathbb{P}:S\neq\varnothing\}$$

such that $\kappa(u)\cap \kappa(v)=\varnothing$ for every edge $uv\in E(G)$. The Kromatic symmetric function is

$\overline{X}_G$0

In the unweighted case one writes simply $\overline{X}_G$1 (Crew et al., 2023).

This definition deforms the ordinary chromatic symmetric function by allowing each vertex to carry a nonempty set of colors rather than a single color. The lowest possible homogeneous degree of $\overline{X}_G$2 is $\overline{X}_G$3, corresponding to singleton color sets, and the degree-$\overline{X}_G$4 component is the classical weighted chromatic symmetric function $\overline{X}_G$5. In particular, for unweighted graphs the lowest-degree part of $\overline{X}_G$6 is $\overline{X}_G$7 (Pierson et al., 25 Aug 2025).

Because vertices may receive arbitrarily large finite color sets, $\overline{X}_G$8 generally has unbounded degree. Accordingly, several papers place it in the completion $\overline{X}_G$9 of the ring of symmetric functions rather than in the ordinary graded ring $\overline{X}_{(G,\omega)}$0 (Lin et al., 19 Sep 2025).

A useful exact relation to classical CSFs is given by clan graphs. For a weighted graph $\overline{X}_{(G,\omega)}$1,

$\overline{X}_{(G,\omega)}$2

where $\overline{X}_{(G,\omega)}$3 is the $\overline{X}_{(G,\omega)}$4-clan graph and $\overline{X}_{(G,\omega)}$5. This identity transfers many structural questions about $\overline{X}_{(G,\omega)}$6 to families of ordinary chromatic symmetric functions (Pierson et al., 25 Aug 2025).

2. Algebraic structure and canonical expansions

A basic $\overline{X}_{(G,\omega)}$7-analogue of the augmented monomial basis is defined by

$\overline{X}_{(G,\omega)}$8

where $\overline{X}_{(G,\omega)}$9 is the weighted complete graph on $X_G$0 vertices with vertex weights equal to the parts of $X_G$1. If $X_G$2 denotes the set of stable set covers of $X_G$3, and $X_G$4 is the partition of the total weights of the stable sets in $X_G$5, then

$X_G$6

Thus the $X_G$7-coefficients are nonnegative integers with a direct interpretation in terms of stable set covers (Crew et al., 2023).

The Kromatic symmetric function is multiplicative over disjoint unions:

$X_G$8

It also interacts naturally with Tsujie’s join product $X_G$9. In the completed join algebra $K$0,

$K$1

and the $K$2-theoretic augmented monomials satisfy

$K$3

This join-multiplicativity is central in later constructions of equal-KSF graph pairs and in recent Hopf-algebraic complement results (Pierson et al., 25 Aug 2025).

For weighted graphs, Crew–Pechenik–Spirkl also established a deletion–contraction-type relation. If $K$4 is a nonedge of $K$5, then with the auxiliary graphs $K$6, $K$7, $K$8, $K$9, and $(G,\omega)$0 defined in the paper, one has

$(G,\omega)$1

This provides a recursive mechanism for computing $(G,\omega)$2 in the $(G,\omega)$3-basis (Crew et al., 2023).

Recent work also studies “Kromatic pseudobases” and “coKromatic pseudobases.” In particular, if $(G,\omega)$4 is connected of total weight $(G,\omega)$5, then $(G,\omega)$6 forms a multiplicative pseudobasis of $(G,\omega)$7; by contrast, in the join algebra $(G,\omega)$8 the analogous coKromatic pseudobases exist if and only if the generators are weighted cliques (Lin et al., 19 Sep 2025).

3. Hopf-algebraic and quasisymmetric frameworks

Marberg gave a linearly compact Hopf-algebraic construction of the Kromatic symmetric function. On the completed Hopf algebra of weighted graphs, the coproduct is

$(G,\omega)$9

and there is a unique LC-Hopf algebra morphism

$\omega:V(G)\to \mathbb{N}$0

This places $\omega:V(G)\to \mathbb{N}$1 in the same universal framework as chromatic and quasisymmetric Hopf invariants (Lin et al., 19 Sep 2025).

A parallel construction in $\omega:V(G)\to \mathbb{N}$2 yields a positive expansion of $\omega:V(G)\to \mathbb{N}$3 into multifundamental quasisymmetric functions:

$\omega:V(G)\to \mathbb{N}$4

The coefficients count pairs $\omega:V(G)\to \mathbb{N}$5 where $\omega:V(G)\to \mathbb{N}$6 is an acyclic multi-orientation and $\omega:V(G)\to \mathbb{N}$7 is a multilinear extension with prescribed descent composition. This expansion is one of the first systematic positivity results for KSF beyond the $\omega:V(G)\to \mathbb{N}$8-basis (Marberg, 2023).

The same framework leads to two quasisymmetric $\omega:V(G)\to \mathbb{N}$9-analogues. For an ordered graph $$\kappa:V(G)\to \{S\subset \mathbb{P}:S\neq\varnothing\}$$0, the first is

$$\kappa:V(G)\to \{S\subset \mathbb{P}:S\neq\varnothing\}$$1

and the second is

$$\kappa:V(G)\to \{S\subset \mathbb{P}:S\neq\varnothing\}$$2

where in the latter case $$\kappa:V(G)\to \{S\subset \mathbb{P}:S\neq\varnothing\}$$3 counts pairwise ascents across the set-valued coloring. Both specialize to $$\kappa:V(G)\to \{S\subset \mathbb{P}:S\neq\varnothing\}$$4 at $$\kappa:V(G)\to \{S\subset \mathbb{P}:S\neq\varnothing\}$$5 (Marberg, 2023).

These two analogues behave differently. The function $$\kappa:V(G)\to \{S\subset \mathbb{P}:S\neq\varnothing\}$$6 is symmetric if and only if $$\kappa:V(G)\to \{S\subset \mathbb{P}:S\neq\varnothing\}$$7 is a cluster graph. By contrast, if $$\kappa:V(G)\to \{S\subset \mathbb{P}:S\neq\varnothing\}$$8 is the incomparability graph of a natural unit interval order, then $$\kappa:V(G)\to \{S\subset \mathbb{P}:S\neq\varnothing\}$$9 is symmetric and admits a positive expansion into symmetric Grothendieck functions:

$\kappa(u)\cap \kappa(v)=\varnothing$0

where the sum ranges over Grothendieck $\kappa(u)\cap \kappa(v)=\varnothing$1-tableaux (Marberg, 2023).

A distinct recent direction concerns complements. In the KSF setting, the map

$\kappa(u)\cap \kappa(v)=\varnothing$2

is a Hopf algebra morphism into the join algebra $\kappa(u)\cap \kappa(v)=\varnothing$3, and there is a single morphism sending the KSFs of all unweighted triangle-free graphs to the KSFs of their complements (Lin et al., 19 Sep 2025). This suggests that complement phenomena are substantially more natural in the completed $\kappa(u)\cap \kappa(v)=\varnothing$4-theoretic setting than in the ordinary ring $\kappa(u)\cap \kappa(v)=\varnothing$5.

4. Power-sum expansions, independence polynomials, and Lyndon heaps

Crew, Pechenik, and Spirkl defined a $\kappa(u)\cap \kappa(v)=\varnothing$6-analogue of the power-sum basis by

$\kappa(u)\cap \kappa(v)=\varnothing$7

For a weighted graph $\kappa(u)\cap \kappa(v)=\varnothing$8, Pierson proved the expansion

$\kappa(u)\cap \kappa(v)=\varnothing$9

where the integers $uv\in E(G)$0 are characterized by

$uv\in E(G)$1

with $uv\in E(G)$2 the weighted independence polynomial of the induced subgraph. This gives an explicit $uv\in E(G)$3-expansion and proves that the coefficients are always integers (Pierson, 2024).

The same paper derived a coefficient formula

$uv\in E(G)$4

for $uv\in E(G)$5. In the unweighted case, the lowest-degree terms of this $uv\in E(G)$6-expansion agree with Stanley’s classical $uv\in E(G)$7-expansion of $uv\in E(G)$8 (Pierson, 2024).

A later paper recast these coefficients combinatorially using Lyndon heaps. It introduced a second $uv\in E(G)$9-power-sum basis

$\overline{X}_G$00

and proved especially clean formulas for $\overline{X}_G$01. Most notably,

$\overline{X}_G$02

counts sets of distinct Lyndon heaps on $\overline{X}_G$03 whose union of supports is $\overline{X}_G$04 and whose sizes are the parts of $\overline{X}_G$05. In this basis, $\overline{X}_G$06 is therefore $\overline{X}_G$07-positive (Pierson, 28 Feb 2025).

The same work proved a structural equivalence:

$\overline{X}_G$08

That is, knowing $\overline{X}_G$09 is equivalent to knowing the multiset of independence polynomials of all induced subgraphs of $\overline{X}_G$10 (Pierson, 28 Feb 2025).

This equivalence shortens and conceptualizes earlier counting results. Pierson had already shown that $\overline{X}_G$11 determines the number of copies in $\overline{X}_G$12 of certain induced subgraphs on $\overline{X}_G$13 and $\overline{X}_G$14 vertices, as well as the number of induced subgraphs isomorphic to each graph consisting of a star plus some number of isolated vertices (Pierson, 2024). The independence-polynomial characterization clarifies why these counts are accessible from KSF.

5. Positivity results and non-lifts of classical conjectures

The foundational positivity theorem for KSF is the $\overline{X}_G$15-theoretic lift of Gasharov’s theorem. If $\overline{X}_G$16 is a claw-free incomparability graph, then

$\overline{X}_G$17

where $\overline{X}_G$18 is the symmetric Grothendieck function. Moreover, the coefficient $\overline{X}_G$19 equals the number of Grothendieck $\overline{X}_G$20-tableaux of shape $\overline{X}_G$21. This gives Grothendieck positivity for the entire class of claw-free incomparability graphs and suggests a $\overline{X}_G$22-theoretic interpretation of Gasharov’s Schur-positivity theorem (Crew et al., 2023).

At the same time, the most direct $\overline{X}_G$23-theoretic analogue of elementary positivity fails even for very small graphs. Crew, Pechenik, and Spirkl defined two $\overline{X}_G$24-analogues of the $\overline{X}_G$25-basis,

$\overline{X}_G$26

and showed that the path $\overline{X}_G$27 is not positive in either basis. Their conclusion is explicit: the Stanley–Stembridge conjecture does not have such a lift to $\overline{X}_G$28-theory and is therefore unlikely to be amenable to a topological perspective of that form (Crew et al., 2023).

This contrast is characteristic of the theory. Grothendieck positivity survives for a substantial graph class, but direct $\overline{X}_G$29-lifts of classical $\overline{X}_G$30-positivity do not. A plausible implication is that the appropriate positivity structures for KSF are fundamentally different from those governing the ordinary chromatic symmetric function.

The quasisymmetric side sharpens this picture. The function $\overline{X}_G$31 has a complete symmetry classification—cluster graphs and only cluster graphs—whereas the other $\overline{X}_G$32-analogue $\overline{X}_G$33 is symmetric and Grothendieck-positive on incomparability graphs of natural unit interval orders (Marberg, 2023). The theory therefore does not support a single universal positivity paradigm; rather, it supports several basis-dependent positivity phenomena with different graph-theoretic domains.

6. Distinguishing power, counterexamples, and current status

A major motivation for KSF was the expectation that it might distinguish more graphs than the classical CSF. This expectation is justified in many concrete cases. The Kromatic symmetric function is known to distinguish some pairs of graphs with the same CSF, and later work showed that many graph pairs arising from constructions of Orellana–Scott and of Aliste-Prieto, Crew, Spirkl, and Zamora that share the same CSF are nevertheless distinguished by KSF (Pierson et al., 25 Aug 2025).

This led Pierson to conjecture that $\overline{X}_G$34 distinguishes all graphs (Pierson, 2024). That conjecture is now false. In 2025, four pairs of nonisomorphic $\overline{X}_G$35-vertex graphs with equal Kromatic symmetric functions were exhibited, and the paper states that these are the smallest counterexamples found by exhaustive computation in Sage (Pierson et al., 25 Aug 2025).

The same paper also gave several closure constructions producing larger equal-KSF pairs from smaller ones. Equality is preserved under disjoint union and under joins:

$\overline{X}_G$36

More elaborate “attach to all but one vertex” and “attach to one vertex” constructions were also established, yielding infinite families of equal-KSF pairs (Pierson et al., 25 Aug 2025).

Even after the global conjecture was disproved, the theory retains substantial distinguishing power. The same paper emphasizes that KSF is stronger than CSF on many constructed families and that it detects induced-subgraph statistics unavailable to ordinary CSF (Pierson et al., 25 Aug 2025). It remains open whether KSF distinguishes all trees, a natural question given that the analogous CSF problem for trees is longstanding (Pierson et al., 25 Aug 2025).

The present state of the subject is therefore mixed but sharply defined. KSF is not a complete invariant for all graphs, yet it refines CSF in several strong directions: it supports stable-set-cover, power-sum, multifundamental, and Grothendieck expansions; it is equivalent to the multiset of independence polynomials of induced subgraphs; and it distinguishes many same-CSF pairs that classical theory cannot separate (Pierson, 28 Feb 2025). This suggests that its long-term significance may lie less in universal graph reconstruction than in the richer algebraic and enumerative structures it makes accessible.