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On graphs with equal and different Kromatic symmetric functions

Laura Pierson, Soham Samanta

TL;DR

The paper investigates the extent to which the Kromatic symmetric function $\overline{X}_G$ strengthens graph distinction beyond Stanley's chromatic symmetric function $X_G$. It provides four 8-vertex nonisomorphic counterexamples with equal KSFs, disproving the conjecture that KSFs distinguish all graphs, and presents constructive methods (disjoint unions, joins, and vertex-attachment) to generate infinite families of equal-KSF pairs. It also assesses prominent CSF-equal families from the Orellana–Scott and Aliste-Prieto–Crew–Spirkl–Zamora constructions, finding that many of these are distinguished by KSFs under natural hypotheses through detailed subgraph-counting arguments. The work combines independence-polynomial expansions, clan-graph/augmented-monomial frameworks, and intricate combinatorial counts to map where KSFs align with or exceed CSFs in distinguishing nonisomorphic graphs, and it highlights avenues for further study via contraction–deletion techniques.

Abstract

The Kromatic symmetric function (KSF) $\overline{X}_G$ of a graph $G$ is a $K$-analogue introduced by Crew, Pechenik, and Spirkl in arXiv:2301.02177 of Stanley's chromatic symmetric function (CSF) $X_G$. The KSF is known to distinguish some pairs of graphs with the same CSF. The first author showed in arXiv:2403.15929 and arXiv:2502.21285 that the number of copies in $G$ of certain induced subgraphs can be determined given $\overline{X}_G$, and conjectured that $\overline{X}_G$ distinguishes all graphs. We disprove that conjecture by finding four pairs of 8-vertex graphs with equal KSF, as well as giving several ways to use existing graph pairs with equal KSF to construct larger graph pairs that also have equal KSF. On the other hand, we show that many of the graph pairs from the constructions of Orellana and Scott in arXiv:1308.6005 and of Aliste-Prieto, Crew, Spirkl, and Zamora in arXiv:2007.11042 of graphs with the same CSF are distinguished by the KSF, thus also giving some new examples of cases where the KSF is a stronger invariant than the CSF.

On graphs with equal and different Kromatic symmetric functions

TL;DR

The paper investigates the extent to which the Kromatic symmetric function strengthens graph distinction beyond Stanley's chromatic symmetric function . It provides four 8-vertex nonisomorphic counterexamples with equal KSFs, disproving the conjecture that KSFs distinguish all graphs, and presents constructive methods (disjoint unions, joins, and vertex-attachment) to generate infinite families of equal-KSF pairs. It also assesses prominent CSF-equal families from the Orellana–Scott and Aliste-Prieto–Crew–Spirkl–Zamora constructions, finding that many of these are distinguished by KSFs under natural hypotheses through detailed subgraph-counting arguments. The work combines independence-polynomial expansions, clan-graph/augmented-monomial frameworks, and intricate combinatorial counts to map where KSFs align with or exceed CSFs in distinguishing nonisomorphic graphs, and it highlights avenues for further study via contraction–deletion techniques.

Abstract

The Kromatic symmetric function (KSF) of a graph is a -analogue introduced by Crew, Pechenik, and Spirkl in arXiv:2301.02177 of Stanley's chromatic symmetric function (CSF) . The KSF is known to distinguish some pairs of graphs with the same CSF. The first author showed in arXiv:2403.15929 and arXiv:2502.21285 that the number of copies in of certain induced subgraphs can be determined given , and conjectured that distinguishes all graphs. We disprove that conjecture by finding four pairs of 8-vertex graphs with equal KSF, as well as giving several ways to use existing graph pairs with equal KSF to construct larger graph pairs that also have equal KSF. On the other hand, we show that many of the graph pairs from the constructions of Orellana and Scott in arXiv:1308.6005 and of Aliste-Prieto, Crew, Spirkl, and Zamora in arXiv:2007.11042 of graphs with the same CSF are distinguished by the KSF, thus also giving some new examples of cases where the KSF is a stronger invariant than the CSF.

Paper Structure

This paper contains 4 sections, 23 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Graphs with equal Kromatic symmetric function
  3. Orellana-Scott construction
  4. Aliste-Prieto--Crew--Spirkl--Zamora construction

Key Result

Theorem 1.3

Knowing $\overline{X}_G$ is equivalent to knowing the multiset of independence polynomials of induced subgraphs of $G$.

Theorems & Definitions (52)

  • Definition 1.1: Stanley stanley1995symmetric; Crew-Spirkl crew2020deletion for weighted version
  • Definition 1.2: Crew-Pechenik-Spirkl crew2023kromatic
  • Theorem 1.3: Pierson new_p_expansion, Corollary 1.6
  • Theorem 1.4: Pierson pierson2024counting, Theorems 3 and 4
  • Conjecture 1.5: Pierson pierson2024counting, Conjecture 2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof : Proof 1.
  • Proposition 2.3: Stanley stanley1995symmetric, Theorem 2.5; Crew-Spirkl crew2020deletion, Lemma 1
  • ...and 42 more