The Total Chromatic Quasisymmetric Functions of a Graph

TL;DR

This paper introduces two new variants of the chromatic quasisymmetric function for graphs, normalizing for vertex labelings and using acyclic orientations, and analyzes their properties and coefficients.

Contribution

It constructs and studies two novel variants of the chromatic quasisymmetric function, addressing labeling dependence and the tree isomorphism conjecture.

Findings

Derived explicit formulas for star graph coefficients.

Provided a combinatorial proof for a binomial identity related to the coefficients.

Compared properties of the two variants' coefficients in the monomial basis.

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.