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Counting induced subgraphs with the Kromatic symmetric function

Laura Pierson

TL;DR

This work studies the Kromatic symmetric function $\overline{X}_G$, a $K$-theoretic analogue of Stanley's chromatic symmetric function, and conjectures that $\overline{X}_G$ distinguishes all graphs. The authors exploit a expansion of $\overline{X}_G$ in the basis $\overline{\widetilde{m}}_\lambda$ and a key lemma that links the coefficients to counts of covers of $V(G)$ by stable sets, enabling the recovery of induced-subgraph counts from $\overline{X}_G$. They prove that $\overline{X}_G$ determines the numbers of induced copies of all graphs on 4 vertices except a small set (which are then resolved via linear equations), all graphs on 5 vertices in a substantial subset, and induced star-related subgraphs of arbitrary size. These results provide strong evidence for the conjecture and yield an alternative means to distinguish certain graph pairs that share the ordinary $X_G$, highlighting the potential of $\overline{X}_G$ to capture finer graph properties, including tree structure.

Abstract

The chromatic symmetric function $X_G$ is a sum of monomials corresponding to proper vertex colorings of a graph $G$. Crew, Pechenik, and Spirkl (2023) recently introduced a $K$-theoretic analogue $\overline{X}_G$ called the Kromatic symmetric function, where each vertex is instead assigned a nonempty set of colors such that adjacent vertices have nonoverlapping color sets. $X_G$ does not distinguish all graphs, but a longstanding open question is whether it distinguishes all trees. We conjecture that $\overline{X}_G$ does distinguish all graphs. As evidence towards this conjecture, we show that $\overline{X}_G$ determines the number of copies in $G$ of certain induced subgraphs on 4 and 5 vertices as well as the number of induced subgraphs isomorphic to each graph consisting of a star plus some number of isolated vertices.

Counting induced subgraphs with the Kromatic symmetric function

TL;DR

This work studies the Kromatic symmetric function , a -theoretic analogue of Stanley's chromatic symmetric function, and conjectures that distinguishes all graphs. The authors exploit a expansion of in the basis and a key lemma that links the coefficients to counts of covers of by stable sets, enabling the recovery of induced-subgraph counts from . They prove that determines the numbers of induced copies of all graphs on 4 vertices except a small set (which are then resolved via linear equations), all graphs on 5 vertices in a substantial subset, and induced star-related subgraphs of arbitrary size. These results provide strong evidence for the conjecture and yield an alternative means to distinguish certain graph pairs that share the ordinary , highlighting the potential of to capture finer graph properties, including tree structure.

Abstract

The chromatic symmetric function is a sum of monomials corresponding to proper vertex colorings of a graph . Crew, Pechenik, and Spirkl (2023) recently introduced a -theoretic analogue called the Kromatic symmetric function, where each vertex is instead assigned a nonempty set of colors such that adjacent vertices have nonoverlapping color sets. does not distinguish all graphs, but a longstanding open question is whether it distinguishes all trees. We conjecture that does distinguish all graphs. As evidence towards this conjecture, we show that determines the number of copies in of certain induced subgraphs on 4 and 5 vertices as well as the number of induced subgraphs isomorphic to each graph consisting of a star plus some number of isolated vertices.
Paper Structure (6 sections, 5 theorems, 27 equations, 1 figure)

This paper contains 6 sections, 5 theorems, 27 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Key lemma
  4. Proof of Theorem \ref{['thm:subgraphs_4']}
  5. Proof of Theorem \ref{['thm:subgraphs_5']}
  6. Proof of Theorem \ref{['thm:stars']}

Key Result

Theorem 3

The number of induced copies in $G$ of the following order 4 graphs can be computed from $\overline{X}_G$: and the counts of the remaining 4 order 4 graphs satisfy a system of 3 linear equations determined by $\overline{X}_G$.

Figures (1)

  • Figure 1: System of linear equations to count induced subgraphs of order 5

Theorems & Definitions (10)

  • Definition 1: Crew, Pechenik, and Spirkl (2023), kromatic2023
  • Conjecture 2
  • Theorem 3
  • Theorem 4
  • Example 5
  • Theorem 6
  • Definition 7: Crew, Pechenik, and Spirkl (2023), kromatic2023
  • Theorem 8: Crew, Pechenik, and Spirkl (2023), kromatic2023
  • Lemma 9
  • proof