Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Total Chromatic Quasisymmetric Functions of a Graph

Laura Colmenarejo, Ian Klein

TL;DR

The paper develops two total variants of the chromatic quasisymmetric function: a labeling-total variant $T\chi_G^L(q)$ and an orientation-total variant $T\chi_G^o(q)$. It establishes foundational properties for both, including disjoint-union behavior, palindromicity, reversal-invariance, and, for the orientation variant, a deletion-near-contraction relation. Focusing on the star graph, it derives explicit $M$-basis formulas and shows how the labeling and orientation viewpoints relate via bijections and symmetry; a key binomial identity underpins the labeling coefficients. The combinatorial model for the binomial identity provides a deep connection between algebraic coefficients and barrier/marked-box configurations. Collectively, the work connects structural graph properties to refined symmetric-quasisymmetric function summaries with potential implications for tree-distinguishing questions and related conjectures.

Abstract

In this paper, we introduce and study two variants of the chromatic quasisymmetric function of a graph: the total chromatic quasisymmetric function via vertex labeling and via acyclic orientations. The original definition of the chromatic quasisymmetric function of a graph by Shareshian and Wachs depends on a labeling of the vertices of the graph, which directly affects the properties of the coefficients appearing in the decomposition of the chromatic quasisymmetric function of a graph into different bases. Motivated by this, we construct the first variant of the chromatic quasisymmetric function of a graph by normalizing it with respect to all the labelings of the vertices. The second variant is motivated by the \emph{tree isomorphism conjecture} and is constructed in terms of acyclic orientations. We investigate the properties of the coefficients in the expansion in the monomial quasisymmetric basis for both variants and provide a comparative analysis. Furthermore, we derive explicit formulas for the coefficients in the monomial decomposition of the two variants for the star graph. For the labeling-based variant, these coefficients arise from a binomial identity for which we provide a combinatorial proof.

The Total Chromatic Quasisymmetric Functions of a Graph

TL;DR

The paper develops two total variants of the chromatic quasisymmetric function: a labeling-total variant and an orientation-total variant . It establishes foundational properties for both, including disjoint-union behavior, palindromicity, reversal-invariance, and, for the orientation variant, a deletion-near-contraction relation. Focusing on the star graph, it derives explicit -basis formulas and shows how the labeling and orientation viewpoints relate via bijections and symmetry; a key binomial identity underpins the labeling coefficients. The combinatorial model for the binomial identity provides a deep connection between algebraic coefficients and barrier/marked-box configurations. Collectively, the work connects structural graph properties to refined symmetric-quasisymmetric function summaries with potential implications for tree-distinguishing questions and related conjectures.

Abstract

In this paper, we introduce and study two variants of the chromatic quasisymmetric function of a graph: the total chromatic quasisymmetric function via vertex labeling and via acyclic orientations. The original definition of the chromatic quasisymmetric function of a graph by Shareshian and Wachs depends on a labeling of the vertices of the graph, which directly affects the properties of the coefficients appearing in the decomposition of the chromatic quasisymmetric function of a graph into different bases. Motivated by this, we construct the first variant of the chromatic quasisymmetric function of a graph by normalizing it with respect to all the labelings of the vertices. The second variant is motivated by the \emph{tree isomorphism conjecture} and is constructed in terms of acyclic orientations. We investigate the properties of the coefficients in the expansion in the monomial quasisymmetric basis for both variants and provide a comparative analysis. Furthermore, we derive explicit formulas for the coefficients in the monomial decomposition of the two variants for the star graph. For the labeling-based variant, these coefficients arise from a binomial identity for which we provide a combinatorial proof.
Paper Structure (22 sections, 46 theorems, 70 equations, 9 figures, 3 tables)

This paper contains 22 sections, 46 theorems, 70 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Compositions and partitions
  4. Symmetric and quasisymmetric functions
  5. Graphs
  6. Chromatic symmetric and quasisymmetric functions
  7. The total chromatic quasisymmetric function of a graph
  8. Via labelings
  9. Via acyclic orientations
  10. Comparison: Labelings vs acyclic orientations
  11. The star graph
  12. The chromatic quasisymmetric function
  13. Total CQSF via acyclic orientations
  14. Total CQSF via labelings
  15. Proof of Theorem \ref{['thm: first step c_alpha']}
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $G$ be a unit interval graph and consider the expansion of the chromatic symmetric function of $G$ in the $e$-basis, $\chi_G (x)= \sum_{\lambda \vdash n}a_\lambda e_\lambda$. Then, $\chi_G$ is $e$-positive; that is, $a_\lambda \in\mathbb{Z}_{\geq 0}$.

Figures (9)

  • Figure 1: All the configurations for $n=6$ and $s=k=2$.
  • Figure 2: Configurations for $n=6$ and $s=k=2$ satisfying the $j$-condition.
  • Figure 3: Configurations for $n=6$ and $s=k=2$ satisfying $t$ of the $i$-conditions.
  • Figure 4: Set of configurations for $n=6$ and $s=k=2$ together with the $3$ barriers starting at $b_0=1$.
  • Figure 5: The relation of the barriers and conditions.
  • ...and 4 more figures

Theorems & Definitions (104)

  • Theorem 1.1: Stanley-Stembridge conjecture hikita2024proofstanleystembridgeconjecturehikita2025qtchromaticsymmetricfunctions
  • Conjecture 1.2
  • Example 1.3
  • Definition
  • Theorem
  • Theorem
  • Conjecture 1.4: Tree isomorphism conjecture
  • Conjecture 1.5
  • Definition
  • Example 2.1
  • ...and 94 more