Chromatic symmetric function

Symmetric function invariant of graphs

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 $G$ : $X_{G}(x_{1},x_{2},\ldots ):=\sum _{\underset {\text{proper}}{\kappa :V\to \mathbb {N} }}x_{\kappa (v_{1})}x_{\kappa (v_{2})}\cdots x_{\kappa (v_{n})}$

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 $6x_{1}x_{2}x_{3}$
For each pair of colors, there are $2$ ways to color $P_{3}$ yielding the terms $x_{i}^{2}x_{j}$ and $x_{i}x_{j}^{2}$ for $i\neq j$

Altogether, the chromatic symmetric function of $P_{3}$ is then given by: $X_{P_{3}}(x_{1},x_{2},x_{3})=6x_{1}x_{2}x_{3}+x_{1}^{2}x_{2}+x_{1}x_{2}^{2}+x_{1}^{2}x_{3}+x_{1}x_{3}^{2}+x_{2}^{2}x_{3}+x_{2}x_{3}^{2}=6m_{(1,1,1)}+m_{(1,2)}$

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 ${\text{type}}(\pi )=\lambda =\langle 1^{r_{1}}2^{r2}\ldots \rangle$ . 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 ${\textstyle X_{G}=\sum _{\lambda \vdash n}c_{\lambda }e_{\lambda }}$ 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 ${\textstyle (3+1)}$ -free poset, then ${\textstyle X_{G}}$ 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 ${\textstyle (3+1)}$ -free poset, then ${\textstyle X_{G}}$ 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: $X_{G}(x_{1},x_{2},\ldots ;t):=\sum _{\underset {\text{proper}}{\kappa :V\to \mathbb {N} }}t^{|asc(\kappa )|}x_{\kappa (v_{1})}\cdots x_{\kappa (v_{n})}$