Chromatic symmetric function

The chromatic symmetric function is a symmetric function invariant of graphs studied in algebraic graph theory, a branch of mathematics. It is the weight generating function for proper graph colorings, and was originally introduced by Richard Stanley as a generalization of the chromatic polynomial of a graph.

Definition
For a finite graph $$G=(V,E)$$ with vertex set $$V=\{v_1,v_2,\ldots, v_n\}$$, a vertex coloring is a function $$\kappa:V\to C$$ where $$C$$ is a set of colors. A vertex coloring is called proper if all adjacent vertices are assigned distinct colors (i.e., $$\{i,j\}\in E \implies \kappa(i)\neq\kappa(j)$$). The chromatic symmetric function denoted $$X_G(x_1,x_2,\ldots)$$ is defined to be the weight generating function of proper vertex colorings of $$G$$: $$X_G(x_1,x_2,\ldots):=\sum_{\underset{\text{proper}}{\kappa:V\to\N}}x_{\kappa(v_1)}x_{\kappa(v_2)}\cdots x_{\kappa(v_n)}$$

Examples
For $$\lambda$$ a partition, let $$m_\lambda$$ be the monomial symmetric polynomial associated to $$\lambda$$.

Example 1: Complete Graphs
Consider the complete graph $$K_n$$ on $$n$$ vertices:


 * There are $$n!$$ ways to color $$K_n$$ with exactly $$n$$ colors yielding the term $$n!x_1\cdots x_n$$
 * Since every pair of vertices in $$K_n$$ is adjacent, it can be properly colored with no fewer than $$n$$ colors.

Thus, $$X_{K_n}(x_1,\ldots,x_n)=n!x_1\cdots x_n = n!m_{(1,\ldots,1)}$$

Example 2: A Path Graph
Consider the path graph $$P_3$$ of length $$3$$:


 * There are $$3!$$ ways to color $$P_3$$ with exactly $$3$$ colors, yielding the term $$6x_1x_2x_3$$
 * For each pair of colors, there are $$2$$ ways to color $$P_3$$ yielding the terms $$x_i^2x_j$$ and $$x_ix_j^2$$ for $$i\neq j$$

Altogether, the chromatic symmetric function of $$P_3$$ is then given by: $$X_{P_3}(x_1,x_2,x_3) = 6x_1x_2x_3 + x_1^2x_2 + x_1x_2^2 + x_1^2x_3 + x_1x_3^2 + x_2^2x_3 + x_2x_3^2 = 6m_{(1,1,1)} + m_{(1,2)}$$

Properties

 * Let $$\chi_G$$ be the chromatic polynomial of $$G$$, so that $$\chi_G(k)$$ is equal to the number of proper vertex colorings of $$G$$ using at most $$k$$ distinct colors. The values of $$\chi_G$$ can then be computed by specializing the chromatic symmetric function, setting the first $$k$$ variables $$x_i$$ equal to $$1$$ and the remaining variables equal to $$0$$: $$X_G(1^k)=X_G(1,\ldots,1,0,0,\ldots)=\chi_G(k)$$
 * If $$G\amalg H$$ is the disjoint union of two graphs, then the chromatic symmetric function for $$G\amalg H$$ can be written as a product of the corresponding functions for $$G$$ and $$H$$: $$X_{G\amalg H}=X_G\cdot X_H$$
 * A stable partition $$\pi$$ of $$G$$ is defined to be a set partition of vertices $$V$$ such that each block of $$\pi$$ is an independent set in $$G$$. The type of a stable partition $$\text{type}(\pi)$$ is the partition consisting of parts equal to the sizes of the connected components of the vertex induced subgraphs. For a partition $$\lambda\vdash n$$, let $$z_\lambda$$ be the number of stable partitions of $$G$$ with $$\text{type}(\pi)=\lambda=\langle1^{r_1}2^{r2}\ldots\rangle$$. Then, $$X_G$$ expands into the augmented monomial symmetric functions, $$\tilde{m}_\lambda:=r_1!r_2!\cdots m_\lambda$$ with coefficients given by the number of stable partitions of $$G$$: $$X_G=\sum_{\lambda\vdash n}z_\lambda \tilde{m}_\lambda$$
 * Let $$p_\lambda$$ be the power-sum symmetric function associated to a partition $$\lambda$$. For $$S\subseteq E$$, let $$\lambda(S)$$ be the partition whose parts are the vertex sizes of the connected components of the edge-induced subgraph of $$G$$ specified by $$S$$. The chromatic symmetric function can be expanded in the power-sum symmetric functions via the following formula: $$X_G=\sum_{S\subseteq E}(-1)^{|S|}p_{\lambda(S)}$$
 * Let $X_G=\sum_{\lambda\vdash n}c_\lambda e_\lambda$ be the expansion of $$X_G$$ in the basis of elementary symmetric functions $$e_\lambda$$. Let $$\text{sink}(G,s)$$ be the number of acyclic orientations on the graph $$G$$ which contain exactly $$s$$ sinks. Then we have the following formula for the number of sinks: $$\text{sink}(G,s)=\sum_{\underset{l(\lambda)=s}{\lambda\vdash n}}c_\lambda$$

Open Problems
There are a number of outstanding questions regarding the chromatic symmetric function which have received substantial attention in the literature surrounding them.

(3+1)-free Conjecture
For a partition $$\lambda$$, let $$e_\lambda$$ be the elementary symmetric function associated to $$\lambda$$.

A partially ordered set $$P$$ is called $$(3+1)$$-free if it does not contain a subposet isomorphic to the direct sum of the $$3$$ element chain and the $$1$$ element chain. The incomparability graph $$\text{inc}(P)$$ of a poset $$P$$ is the graph with vertices given by the elements of $$P$$ which includes an edge between two vertices if and only if their corresponding elements in $$P$$ are incomparable.

Conjecture (Stanley-Stembridge) Let $$G$$ be the incomparability graph of a $(3+1)$ -free poset, then $X_G$ is $$e$$-positive.

A weaker positivity result is known for the case of expansions into the basis of Schur functions.

Theorem (Gasharov) Let $$G$$ be the incomparability graph of a $(3+1)$ -free poset, then $X_G$ is $$s$$-positive.

In the proof of the theorem above, there is a combinatorial formula for the coefficients of the Schur expansion given in terms of $$P$$-tableaux which are a generalization of semistandard Young tableaux instead labelled with the elements of $$P$$.

Generalizations
There are a number of generalizations of the chromatic symmetric function:


 * There is a categorification of the invariant into a homology theory which is called chromatic symmetric homology. This homology theory is known to be a stronger invariant than the chromatic symmetric function alone. The chromatic symmetric function can also be defined for vertex-weighted graphs, where it satisfies a deletion-contraction property analogous to that of the chromatic polynomial. If the theory of chromatic symmetric homology is generalized to vertex-weighted graphs as well, this deletion-contraction property lifts to a long exact sequence of the corresponding homology theory.
 * There is also a quasisymmetric refinement of the chromatic symmetric function which can be used to refine the formulae expressing $$X_G$$ in terms of Gessel's basis of fundamental quasisymmetric functions and the expansion in the basis of Schur functions. Fixing an order for the set of vertices, the ascent set of a proper coloring $$\kappa$$ is defined to be $$\text{asc}(\kappa)=\{\{i,j\}\in E:i<j \text{ and } \kappa(i)<\kappa(j)\}$$. The chromatic quasisymmetric function of a graph $$G$$ is then defined to be: $$X_G(x_1,x_2,\ldots;t):=\sum_{\underset{\text{proper}}{\kappa:V\to \N}}t^{|asc(\kappa)|}x_{\kappa(v_1)}\cdots x_{\kappa(v_n)}$$