Related papers: Chromatic Signed-Symmetric Functions of Signed Gra…

Stanley [9] introduced the chromatic symmetric function ${\bf X}_G$ associated to a simple graph $G$ as a generalization of the chromatic polynomial of $G$. In this paper we present a novel technique to write ${\bf X}_G$ as a linear…

In 1995, Richard Stanley introduced the chromatic symmetric function $X_G$ of a graph $G$ and proved that, when written in terms of the elementary symmetric functions, it reveals the number of acyclic orientations of $G$ with a given number…

A well-known open problem in graph theory asks whether Stanley's chromatic symmetric function, a generalization of the chromatic polynomial of a graph, distinguishes between any two non-isomorphic trees. Previous work has proven the…

Stanley associated with a graph G a symmetric function X_G which reduces to G's chromatic polynomial under a certain specialization of variables. He then proved various theorems generalizing results about the chromatic polynomial, as well…

In 1995, Stanley introduced the chromatic symmetric function of a graph, which specializes to its chromatic polynomial, and which has been the focus of intense research. In 2017, Shareshian, Wachs, and Ellzey defined a refinement of this…

Stanley introduced the concept of chromatic symmetric functions of graphs which extends and refines the notion of chromatic polynomials of graphs, and asked whether trees are determined up to isomorphism by their chromatic symmetric…

Richard P. Stanley defined the chromatic symmetric function of a simple graph and has conjectured that every tree is determined by its chromatic symmetric function. Recently, Takahiro Hasebe and the author proved that the order…

Stanley's symmetrized chromatic polynomial is a generalization of the ordinary chromatic polynomial to a graph invariant with values in a ring of polynomials in infnitely many variables. The ordinary chromatic polynomial is a specialization…

We introduce $H$-chromatic symmetric functions, $X_{G}^{H}$, which use the $H$-coloring of a graph $G$ to define a generalization of Stanley's chromatic symmetric functions. We say two graphs $G_1$ and $G_2$ are $H$-chromatically equivalent…

Stanley asked whether a tree is determined up to isomorphism by its chromatic symmetric function. We approach Stanley's problem by studying the relationship between the chromatic symmetric function and other invariants. First, we prove…

Let G be a graph, and let $\chi$G be its chromatic polynomial. For any non-negative integers i, j, we give an interpretation for the evaluation $\chi$ (i) G (--j) in terms of acyclic orientations. This recovers the classical interpretations…

The chromatic symmetric function of a graph is a generalization of the chromatic polynomial. The key motivation for studying the structure of a chromatic symmetric function is to answer positivity conjectures by Stanley in 1995 and Gasharov…

Stanley defined the chromatic symmetric function of a graph, and Shareshian and Wachs introduced a refinement, namely the chromatic quasisymmetric function of a labeled graph. In this paper, we define the chromatic quasisymmetric function…

This article is dedicated to the study of positivity phenomena for the chromatic symmetric function of a graph with respect to various bases of symmetric functions. We give a new proof of Gasharov's theorem on the Schur-positivity of the…

Stanley has studied a symmetric function generalization X_G of the chromatic polynomial of a graph G. The innocent-looking Stanley-Stembridge Poset Chain Conjecture states that the expansion of X_G in terms of elementary symmetric functions…

Tatsuyuki Hikita recently proved the Stanley--Stembridge conjecture using probabilistic methods, showing that the chromatic symmetric functions of unit interval graphs are $e$-positive. Finding a combinatorial interpretation for these…

The chromatic symmetric function $X_G$ of a graph $G$ was introduced by Stanley. In this paper we introduce a quasisymmetric generalization $X^k_G$ called the $k$-chromatic quasisymmetric function of $G$ and show that it is positive in the…

In this paper, we study positivity phenomena for the $e$-coefficients of Stanley's chromatic function of a graph. We introduce a new combinatorial object: the {\em correct} sequences of unit interval orders, and using these, in certain…

Chromatic quasisymmetric functions of labeled graphs were defined by Shareshian and Wachs as a refinement of Stanley's chromatic symmetric functions. In this extended abstract, we consider an extension of their definition from labeled…

Let $T$ be an unrooted tree. The \emph{chromatic symmetric function} $X_T$, introduced by Stanley, is a sum of monomial symmetric functions corresponding to proper colorings of $T$. The \emph{subtree polynomial} $S_T$, first considered…