Marked Chromatic Polynomials

• Marked chromatic polynomials are refined graph-coloring invariants that supplement proper colorings with additional marking data attached to vertices, edges, or hyperedges.
• They employ q-analogue techniques and deletion–contraction recurrences to establish polynomiality and uncover rich combinatorial structures like reciprocity and order polytope relations.
• These invariants bridge classical chromatic polynomial theory with chromatic symmetric functions, enhancing discrimination in graphs, hypergraphs, and subspace arrangements.

Marked chromatic polynomials are refinements of chromatic counting in which proper colorings are supplemented by additional marking data attached to vertices, colors, hyperedges, or related combinatorial structures. In current usage, the term covers several closely related constructions rather than a single canonical invariant. Among the most developed versions are the graph-theoretic $q$-chromatic polynomial $\chi_G^\lambda(q,n)$, where a vertex weight vector $\lambda$ records the statistic $\sum_v \lambda_v c(v)$; marked multi-coloring enumerators for hypergraphs and subspace arrangements; and marked graph polynomials whose deletion–contraction specializations recover chromatic-symmetric-function information (Bajo et al., 2024, P et al., 28 Jul 2025, Aliste-Prieto et al., 2022).

1. Terminological scope and basic variants

The literature uses “marked chromatic polynomial” for several allied refinements of ordinary chromatic polynomials. In the graph setting, the marking can be a fixed integer vector $\lambda\in\mathbb Z^V$, viewed as a linear form on vertex colors; in the hypergraph setting, it can be a choice of special vertices together with multiplicity data $\mathbf m$; in multivariate hypergraph theory, it can mean that some colors are designated as primary and that hyperedges becoming primary are tracked by edge variables $t_e$; and in marked-graph theory, each vertex carries a mark $(w,d)$ that evolves under contraction (Bajo et al., 2024, P et al., 28 Jul 2025, White, 2010, Aliste-Prieto et al., 2022).

Framework	Marking data	Counted or encoded object
$q$-chromatic polynomial of a graph	$\lambda\in\mathbb Z^V$	Proper colorings weighted by $\chi_G^\lambda(q,n)$0
Marked chromatic polynomial of a hypergraph	Special vertices and multiplicities $\chi_G^\lambda(q,n)$1	Marked multi-colorings
Multivariate chromatic polynomial of a hypergraph	Primary colors and edge variables $\chi_G^\lambda(q,n)$2	Colorings weighted by primary monochromatic hyperedges
$\chi_G^\lambda(q,n)$3- and $\chi_G^\lambda(q,n)$4-polynomials of marked graphs	Vertex marks $\chi_G^\lambda(q,n)$5	Deletion–contraction invariants linked to $\chi_G^\lambda(q,n)$6
Bivariate chromatic polynomials of signed graphs	Paired and unpaired colors	Proper signed colorings with a distinguished color subset

These frameworks share a common principle: chromatic enumeration is refined by an auxiliary statistic that is preserved by a recurrence, a generating series, or a specialization. They differ, however, in what is being marked. In some papers the marking is vertex-based, in others it is color-based, hyperedge-based, or encoded by contraction data. This suggests that “marked chromatic polynomial” functions more as a family resemblance than as a unique formal definition.

2. The graph-theoretic $\chi_G^\lambda(q,n)$7-chromatic polynomial

For a graph $\chi_G^\lambda(q,n)$8 and a fixed integer vector $\chi_G^\lambda(q,n)$9, the $\lambda$0-weighted coloring enumerator is defined by

$\lambda$1

Here $\lambda$2, and the exponent records a linear statistic of the coloring. When $\lambda$3, the weight disappears and one recovers the ordinary chromatic polynomial: $\lambda$4 In this sense, the construction is a genuine $\lambda$5-analogue of chromatic counting, and $\lambda$6 plays the role of a marking or weighting of the vertices (Bajo et al., 2024).

A central theorem states that $\lambda$7 is a polynomial in the $\lambda$8-integer

$\lambda$9

Precisely, there exists a unique polynomial

$\sum_v \lambda_v c(v)$0

such that

$\sum_v \lambda_v c(v)$1

This $\sum_v \lambda_v c(v)$2 is called the $\sum_v \lambda_v c(v)$3-chromatic polynomial. The result places the marked enumeration in a polynomial framework parallel to the ordinary chromatic polynomial, but with $\sum_v \lambda_v c(v)$4 replacing $\sum_v \lambda_v c(v)$5.

The special case $\sum_v \lambda_v c(v)$6 identifies the construction with the principal evaluation of Stanley’s chromatic symmetric function: $\sum_v \lambda_v c(v)$7 This relation makes the $\sum_v \lambda_v c(v)$8-chromatic polynomial a bridge between weighted graph coloring and chromatic symmetric function theory. It also explains why the invariant is especially relevant in the study of trees and other graph families where $\sum_v \lambda_v c(v)$9 is expected to be highly discriminating.

3. Structural theory: polynomiality, recurrences, reciprocity, and order polytopes

The proof of polynomiality proceeds through $\lambda\in\mathbb Z^V$0-Ehrhart theory. Proper colorings of $\lambda\in\mathbb Z^V$1 are partitioned by acyclic orientations, and each region is identified with an open order polytope. The key identity is

$\lambda\in\mathbb Z^V$2

where $\lambda\in\mathbb Z^V$3 denotes the $\lambda\in\mathbb Z^V$4-Ehrhart polynomial

$\lambda\in\mathbb Z^V$5

Chapoton’s theorem on $\lambda\in\mathbb Z^V$6-Ehrhart polynomials then yields polynomiality in $\lambda\in\mathbb Z^V$7 (Bajo et al., 2024).

The $\lambda\in\mathbb Z^V$8-chromatic polynomial satisfies a deletion–contraction relation analogous to the classical one, but contraction modifies the markings. If $\lambda\in\mathbb Z^V$9, then

$\mathbf m$0

Thus the combinatorics of contraction is coupled to addition of vertex weights. By iterating a splitting trick, any $\mathbf m$1 with nonnegative $\mathbf m$2 can be expressed as an integer linear combination of invariants with $\mathbf m$3: $\mathbf m$4

The theory also includes an integrality-type statement. Writing $\mathbf m$5, one has

$\mathbf m$6

with coefficients that are polynomials in $\mathbf m$7. A second structural formula expresses $\mathbf m$8 through the Möbius function of the lattice of flats of the graphical arrangement, and for trees the formula simplifies because $\mathbf m$9. These results show that the invariant admits basis expansions with well-controlled coefficients after multiplication by $t_e$0.

There is also a $t_e$1-reciprocity theorem paralleling Stanley’s reciprocity for $t_e$2. For $t_e$3,

$t_e$4

where the sum ranges over pairs consisting of an $t_e$5-coloring $t_e$6 and a compatible acyclic orientation $t_e$7. This is a weighted count of compatible colorings and orientations, and it makes the reciprocity phenomenon visible at the marked level.

4. Hypergraph and subspace-arrangement marked chromatic polynomials

For a hypergraph $t_e$8 together with a chosen subset of special vertices $t_e$9, and multiplicity vector $(w,d)$0, a marked multi-coloring assigns to each vertex a set of colors from $(w,d)$1, or a multiset if the vertex is special, of total size $(w,d)$2, subject to the condition that for every edge $(w,d)$3,

$(w,d)$4

The number of such colorings is denoted $(w,d)$5, and it is called the marked chromatic polynomial of $(w,d)$6 associated to $(w,d)$7 (P et al., 28 Jul 2025).

The associated generating function is the marked independence series. Its defining combinatorics allows vertices outside the special set to appear at most once, while special vertices may repeat. The main coefficient theorem identifies marked chromatic polynomials as coefficients of a $(w,d)$8-th power: $(w,d)$9 Equivalently, the coefficient of $q$0 in $q$1 is $q$2. This makes polynomiality in $q$3 immediate from the binomial-series expansion of $q$4.

The same framework extends to subspace arrangements. For a finite arrangement $q$5 and a set of special nodes $q$6, an $q$7-marked $q$8-coloring associated to $q$9 is a choice of color sets $\lambda\in\mathbb Z^V$0, with $\lambda\in\mathbb Z^V$1, where $\lambda\in\mathbb Z^V$2 is a multiset if $\lambda\in\mathbb Z^V$3, such that no tuple of chosen colors lies in any $\lambda\in\mathbb Z^V$4. The number of such colorings, denoted $\lambda\in\mathbb Z^V$5, is proved to be a polynomial in $\lambda\in\mathbb Z^V$6. The count is expressed as a sum of characteristic polynomials of derived arrangements, so polynomiality comes from the characteristic-polynomial theory of arrangements.

A further consequence concerns non-negativity. For a hyperplane arrangement,

$\lambda\in\mathbb Z^V$7

This generalizes the graph case

$\lambda\in\mathbb Z^V$8

For general simple hypergraphs, the paper proves a necessary condition: if $\lambda\in\mathbb Z^V$9, then every edge of $\chi_G^\lambda(q,n)$00 must have even cardinality. It then conjectures that

$\chi_G^\lambda(q,n)$01

5. Multivariate and marked deletion–contraction formalisms

A broader hypergraph framework is given by the multivariate chromatic polynomial

$\chi_G^\lambda(q,n)$02

where colors $\chi_G^\lambda(q,n)$03 are designated as primary, and $\chi_G^\lambda(q,n)$04 is the set of hyperedges that are monochromatic in a primary color. The variables $\chi_G^\lambda(q,n)$05 record exactly which hyperedges become primary under the coloring. This polynomial generalizes the bivariate chromatic polynomial of Dohmen–Pönitz–Tittmann and the multivariate coboundary polynomial, and it is explicitly presented as a marked-coloring framework (White, 2010).

The polynomial satisfies a deletion–contraction–extraction recurrence: $\chi_G^\lambda(q,n)$06 with multiplicativity on disjoint unions. It also admits a Möbius-inversion description on the lattice of connected partitions of the hypergraph. In this setting, “marking” is carried by the distinction between primary and non-primary colors together with the edge variables.

A different marked formalism arises from marked graphs. A mark is a pair

$\chi_G^\lambda(q,n)$07

with semigroup law

$\chi_G^\lambda(q,n)$08

A marked graph is a graph together with a mark on each vertex. Its $\chi_G^\lambda(q,n)$09-polynomial is defined recursively by deletion–contraction, with contraction replacing two endpoint marks by their dot-sum. The $\chi_G^\lambda(q,n)$10-polynomial is obtained from $\chi_G^\lambda(q,n)$11 by the undotting substitution

$\chi_G^\lambda(q,n)$12

This framework generalizes the $\chi_G^\lambda(q,n)$13-polynomial of Noble and Welsh and is a specialization of the $\chi_G^\lambda(q,n)$14-polynomial of Ellis-Monaghan and Moffatt (Aliste-Prieto et al., 2022).

Signed-graph theory adds a related bivariate refinement. For a signed graph $\chi_G^\lambda(q,n)$15, the pair

$\chi_G^\lambda(q,n)$16

counts proper colorings from $\chi_G^\lambda(q,n)$17-color sets, where $\chi_G^\lambda(q,n)$18 records the number of unpaired colors. The specialization

$\chi_G^\lambda(q,n)$19

recovers the ordinary signed chromatic quasipolynomial, while

$\chi_G^\lambda(q,n)$20

recovers the ordinary chromatic polynomial of the underlying graph. The paper explicitly situates this construction near the marked/generalized chromatic-polynomial literature and derives dominating-vertex deletion formulae tailored to the signed setting (Greaves et al., 2024).

6. Chromatic symmetric functions, tree distinction, and current limits

The strongest current interaction between marked chromatic polynomials and chromatic symmetric functions occurs in the $\chi_G^\lambda(q,n)$21-chromatic theory of graphs. For a graph $\chi_G^\lambda(q,n)$22, the paper introduces $\chi_G^\lambda(q,n)$23-partitions: an $\chi_G^\lambda(q,n)$24-tuple $\chi_G^\lambda(q,n)$25 with $\chi_G^\lambda(q,n)$26 and $\chi_G^\lambda(q,n)$27 whenever $\chi_G^\lambda(q,n)$28. If $\chi_G^\lambda(q,n)$29 counts them, then

$\chi_G^\lambda(q,n)$30

where $\chi_G^\lambda(q,n)$31 runs over acyclic orientations and linear extensions of the induced posets. For a tree $\chi_G^\lambda(q,n)$32,

$\chi_G^\lambda(q,n)$33

Thus the leading coefficient of the $\chi_G^\lambda(q,n)$34-chromatic polynomial is, up to a simple factor, the stable principal specialization of the chromatic symmetric function (Bajo et al., 2024).

This leads to a strengthened version of Stanley’s conjecture: $\chi_G^\lambda(q,n)$35 The paper verifies this for all trees up to $\chi_G^\lambda(q,n)$36 vertices and notes that it implies a conjecture that $\chi_G^\lambda(q,n)$37-partition functions distinguish trees. In a parallel marked-graph framework, the $\chi_G^\lambda(q,n)$38-polynomial is tied to the star-basis expansion of the chromatic symmetric function by

$\chi_G^\lambda(q,n)$39

where $\chi_G^\lambda(q,n)$40 is the core of the marked version of $\chi_G^\lambda(q,n)$41. This machinery yields the result that proper trees of diameter at most $\chi_G^\lambda(q,n)$42 can be reconstructed from the chromatic symmetric function (Aliste-Prieto et al., 2022).

At the same time, several papers emphasize that refined chromatic invariants are not automatically complete. In signed-graph theory there exist non-switching-isomorphic signed graphs with a common underlying graph and common chromatic polynomials, so even bivariate refinements have limits as classification tools (Greaves et al., 2024). In nearby categorification work, refined state sums and impropriety polynomials $\chi_G^\lambda(q,n)$43 mark the number of improper edges, but the paper explicitly states that it does not define marked chromatic polynomials in that exact sense (Kauffman, 24 Dec 2025). Similarly, chromatic polynomials of $\chi_G^\lambda(q,n)$44-edge-coloured graphs are described as related in spirit to marked chromatic polynomials, while not reducible to a naive edge-marking model (Beaton et al., 2020).

Taken together, these developments show that marked chromatic polynomials form a rapidly expanding cluster of ideas centered on refined coloring enumeration. Their main structural themes are polynomiality, deletion–contraction-type recurrences, generating-series interpretations, reciprocity, and specialization to chromatic symmetric functions. Their main unresolved theme is discriminating power: the most promising current evidence concerns trees, while comparison results in signed and other generalized settings show that no single chromatic refinement should be expected to be universally complete.