The Total Chromatic Quasisymmetric Functions of a Graph

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.