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The Chromatic Symmetric Function for Unicyclic Graphs

Aram Bingham, Lisa Johnston, Colin Lawson, Rosa Orellana, Jianping Pan, Chelsea Sato

TL;DR

This work analyzes the chromatic symmetric function $\mathbf{X}_G$ for connected unicyclic graphs to determine which structural properties are recoverable from the CSF and how the star-basis decomposition and deletion-near-contraction (DNC) relations reveal cycle size, leaves, and attached trees. It develops explicit hook-coefficient formulas, characterizes the leading partition and leading coefficient, and applies these tools to distinguish subfamilies such as cuttlefish graphs, as well as to address bicyclic graphs via cycle-size and girth arguments. The main contributions include a general hook-coefficient formula $c_{\lambda}$ for unicyclic graphs, a complete description of the leading partition for different $r$ (non-trivial rooted trees) regimes, leading-coefficient formulas conditional on sprout structure, and explicit CSF-based certificates distinguishing cuttlefish graphs and recovering cycle data in bicyclic graphs. Computational data illustrate both the reconstructibility limits of $\mathbf{X}_G$ and the effectiveness of the theoretical framework in identifying cycle sizes and certain degree-related invariants, while also highlighting non-uniqueness phenomena within these graph classes.

Abstract

Motivated by the question of which structural properties of a graph can be recovered from the chromatic symmetric function (CSF), we study the CSF of connected unicyclic graphs. While it is known that there can be non-isomorphic unicyclic graphs with the same CSF, we find experimentally that such examples are rare for graphs with up to 17 vertices. In fact, in many cases we can recover data such as the number of leaves, number of internal edges, cycle size, and number of attached non-trivial trees, by extending known results for trees to unicyclic graphs. These results are obtained by analyzing the CSF of a connected unicyclic graph in the $\textit{star-basis}$ using the deletion-near-contraction (DNC) relation developed by Aliste-Prieto, Orellana and Zamora, and computing the "leading" partition, its coefficient, as well as coefficients indexed by hook partitions. We also give explicit formulas for star-expansions of several classes of graphs, developing methods for extracting coefficients using structural properties of the graph.

The Chromatic Symmetric Function for Unicyclic Graphs

TL;DR

This work analyzes the chromatic symmetric function for connected unicyclic graphs to determine which structural properties are recoverable from the CSF and how the star-basis decomposition and deletion-near-contraction (DNC) relations reveal cycle size, leaves, and attached trees. It develops explicit hook-coefficient formulas, characterizes the leading partition and leading coefficient, and applies these tools to distinguish subfamilies such as cuttlefish graphs, as well as to address bicyclic graphs via cycle-size and girth arguments. The main contributions include a general hook-coefficient formula for unicyclic graphs, a complete description of the leading partition for different (non-trivial rooted trees) regimes, leading-coefficient formulas conditional on sprout structure, and explicit CSF-based certificates distinguishing cuttlefish graphs and recovering cycle data in bicyclic graphs. Computational data illustrate both the reconstructibility limits of and the effectiveness of the theoretical framework in identifying cycle sizes and certain degree-related invariants, while also highlighting non-uniqueness phenomena within these graph classes.

Abstract

Motivated by the question of which structural properties of a graph can be recovered from the chromatic symmetric function (CSF), we study the CSF of connected unicyclic graphs. While it is known that there can be non-isomorphic unicyclic graphs with the same CSF, we find experimentally that such examples are rare for graphs with up to 17 vertices. In fact, in many cases we can recover data such as the number of leaves, number of internal edges, cycle size, and number of attached non-trivial trees, by extending known results for trees to unicyclic graphs. These results are obtained by analyzing the CSF of a connected unicyclic graph in the using the deletion-near-contraction (DNC) relation developed by Aliste-Prieto, Orellana and Zamora, and computing the "leading" partition, its coefficient, as well as coefficients indexed by hook partitions. We also give explicit formulas for star-expansions of several classes of graphs, developing methods for extracting coefficients using structural properties of the graph.
Paper Structure (16 sections, 38 theorems, 102 equations, 16 figures, 1 algorithm)

This paper contains 16 sections, 38 theorems, 102 equations, 16 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Definitions and preliminaries
  3. Graph theory
  4. Symmetric functions
  5. Deletion-near-contraction relation
  6. Coefficients for hook partitions
  7. Paths, cycles, and pans
  8. Leading partitions for unicyclic graphs
  9. One non-trivial rooted tree ($r=1$)
  10. More than one non-trivial rooted tree ($r\geq 2$)
  11. Application to cuttlefish graphs
  12. Leading coefficients for unicylic graphs
  13. One non-trivial rooted tree ($r=1$)
  14. More than one non-trivial rooted tree ($r\geq 2$)
  15. Bicyclic graphs
  16. ...and 1 more sections

Key Result

Theorem 2.2

For any positive integer $k$, let $G_k$ denote a connected graph with $k$ vertices and let $\{G_k\}_{k \ge 1}$ be a family of such graphs. Given a partition $\lambda \vdash n$, of length $\ell$, define $G_{\lambda}=G_{\lambda_1} \sqcup G_{\lambda_2} \sqcup \cdots \sqcup G_{\lambda_{\ell}}$. Then $\{

Figures (16)

  • Figure 1: A connected unicyclic graph with a $c$-cycle and $c$ (potentially trivial) rooted trees.
  • Figure 2: Left: A unicyclic graph $G$ on $14$ vertices; right: $G \setminus I(G)$.
  • Figure 3: The DNC-tree $\mathcal{T}(G)$.
  • Figure 4: A unicyclic graph $G$ on $8$ vertices.
  • Figure 5: Two graphs with the same hook coefficients: $c_{(8)}=3$, $c_{(7,1)}=-9$, $c_{(6,1,1)}=9$, $c_{(5,1^3)}=-3$, and $c_\lambda=0$ for all other hooks $\lambda$.
  • ...and 11 more figures

Theorems & Definitions (82)

  • Example 2.1
  • Theorem 2.2: CVW16
  • Definition 2.3
  • Lemma 2.4
  • Proposition 2.5: AMOZ23
  • Theorem 2.6: AMOZ23
  • Example 2.7
  • Proposition 3.1: gonzalez2024chromatic, Proposition 3.8
  • Lemma 3.2
  • proof
  • ...and 72 more