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All cycle-chords are $e$-positive

David G. L. Wang

TL;DR

This work tackles the problem of $e$-positivity for chromatic symmetric functions of cycle-chord graphs, extending beyond unit-interval graphs. It adopts the composition method developed by Zhou and the author to derive an explicit $e$-expansion for cycle-chords, proving $e$-positivity when $a,b\ge 2$ by showing $X_{\mathrm{CC}_{ab}}=\sum_{I\vDash n}\Delta_I(b)\,w_I\,e_I$ with nonnegative $\Delta_I(b)$. A combinatorial interpretation of the $e$-coefficients is provided via interval dissections, together with an involution that ensures positivity, and leading coefficient formulas $[e_n]X_G=abn$ and $[e_{(n-1)1}]X_G=(a-1)(b-1)(n-2)$ are established. The paper also proposes an $e$-positivity conjecture for theta graphs $\theta_{abc}$ (for $a\ge b\ge c\ge 1$), with $c=1$ and $c=2$ cases previously resolved, linking cycle-chord results to a broader class of non-unit-interval graphs and offering a direction for future combinatorial interpretations.

Abstract

We establish the $e$-positivity of cycle-chord graphs by using the composition method which is developed by Zhou and the author recently. Our method is simpler than the $(e)$-positivity approach which is used for handling cycle-chords with girth at most $4$. We also provide a combinatorial interpretation of the $e$-coefficients, and conjecture that theta graphs are $e$-positive.

All cycle-chords are $e$-positive

TL;DR

This work tackles the problem of -positivity for chromatic symmetric functions of cycle-chord graphs, extending beyond unit-interval graphs. It adopts the composition method developed by Zhou and the author to derive an explicit -expansion for cycle-chords, proving -positivity when by showing with nonnegative . A combinatorial interpretation of the -coefficients is provided via interval dissections, together with an involution that ensures positivity, and leading coefficient formulas and are established. The paper also proposes an -positivity conjecture for theta graphs (for ), with and cases previously resolved, linking cycle-chord results to a broader class of non-unit-interval graphs and offering a direction for future combinatorial interpretations.

Abstract

We establish the -positivity of cycle-chord graphs by using the composition method which is developed by Zhou and the author recently. Our method is simpler than the -positivity approach which is used for handling cycle-chords with girth at most . We also provide a combinatorial interpretation of the -coefficients, and conjecture that theta graphs are -positive.
Paper Structure (4 sections, 10 theorems, 33 equations, 3 figures)

This paper contains 4 sections, 10 theorems, 33 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Establishing the $e$-positivity of cycle-chords
  4. The $e$-positivity conjecture for theta graphs

Key Result

Proposition 2.1

Let $G=(V,E)$ be a graph which contains $3$ vertices with no edges among. Denote by $e_1$, $e_2$ and $e_3$ the edges to be used to link these vertices together. For any set $S\subseteq \{1,2,3\}$, denote by $G_S$ the graph with vertex set $V$ and edge set $E\cup\{e_j\colon j\in S\}$. Then

Figures (3)

  • Figure 1: The tadpole $C_m^l$ and the cycle-chord $\mathrm{CC}_{ab}$.
  • Figure 2: When $i_1\le i_p-s$, we have $\Delta_I(b)=s(i_p-s-i_1)$.
  • Figure 3: When $i_1>i_p-s$, we have $\Delta_I(b)=e_2(i_p-s,\,i_{p+1},\,\dots,\,i_q,\,t)$.

Theorems & Definitions (16)

  • Proposition 2.1: OS14
  • Proposition 2.2: SW16
  • Proposition 2.3: Ell17
  • Lemma 2.4: WZ24X
  • Theorem 2.5: WZ24X
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • ...and 6 more