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Linear relations on star coefficients of the chromatic symmetric function

Rosa Orellana, Foster Tom

Abstract

We prove that the coefficient of the star $\mathfrak{st}_{21^{n-2}}$ in the chromatic symmetric function $X_G$ determines whether a connected graph $G$ is $2$-connected. We also prove new linear relations on other star coefficients of chromatic symmetric functions. This allows us to find new bases for certain spans of chromatic symmetric functions. Finally, we relate the coefficient of the star $\mathfrak{st}_n$ to acyclic orientations.

Linear relations on star coefficients of the chromatic symmetric function

Abstract

We prove that the coefficient of the star in the chromatic symmetric function determines whether a connected graph is -connected. We also prove new linear relations on other star coefficients of chromatic symmetric functions. This allows us to find new bases for certain spans of chromatic symmetric functions. Finally, we relate the coefficient of the star to acyclic orientations.
Paper Structure (6 sections, 34 theorems, 44 equations, 2 figures)

This paper contains 6 sections, 34 theorems, 44 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. The coefficient of $\mathfrak{st}_{21^{n-2}}$ and $2$-connectivity
  4. Bases for spans of chromatic symmetric functions
  5. The coefficient of $\mathfrak{st}_n$ and acyclic orientations
  6. Acknowledgments

Key Result

Theorem 1.2

Let $G$ be a connected graph with $n\geq 3$ vertices and let $X_G=\sum_\lambda c_\lambda\mathfrak{st}_\lambda$. Then the coefficient $c_{21^{n-2}}$ is nonzero if and only if $G$ is $2$-connected.

Figures (2)

  • Figure 1: A deletion-near-contraction tree for a graph $G$
  • Figure 2: Graphs $G$ and $H$ with $X_G=X_H$, but $\kappa(G)=3$ and $\kappa(H)=2$

Theorems & Definitions (69)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Remark 3.4
  • ...and 59 more