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Adjacent cycle-chains are $e$-positive

Foster Tom, Aarush Vailaya

Abstract

We describe a way to decompose the chromatic symmetric function as a positive sum of smaller pieces. We show that these pieces are $e$-positive for cycles. Then we prove that attaching a cycle to a graph preserves the $e$-positivity of these pieces. From this, we prove an $e$-positive formula for graphs of cycles connected at adjacent vertices. We extend these results to graphs formed by connecting a sequence of cycles and cliques.

Adjacent cycle-chains are $e$-positive

Abstract

We describe a way to decompose the chromatic symmetric function as a positive sum of smaller pieces. We show that these pieces are -positive for cycles. Then we prove that attaching a cycle to a graph preserves the -positivity of these pieces. From this, we prove an -positive formula for graphs of cycles connected at adjacent vertices. We extend these results to graphs formed by connecting a sequence of cycles and cliques.

Paper Structure

This paper contains 24 sections, 7 theorems, 65 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Forest Triples
  4. Sign-Reversing Involutions
  5. Cycle Graphs
  6. Step 1:
  7. Step 2:
  8. Step 3:
  9. Counting Fixed Points:
  10. Adjacent Cycle Chains
  11. Cycle Plus Tree Involution
  12. Step 1:
  13. Step 2:
  14. Step 3:
  15. Step 4:
  16. ...and 9 more sections

Key Result

Theorem 3.1

The chromatic symmetric function of any graph $G$ is

Figures (12)

  • Figure 1: All forest triples for $P_2$.
  • Figure 2: Examples of $\mathop{\mathrm{join}}\nolimits$ and $\mathop{\mathrm{break}}\nolimits$ for $\mathop{\mathrm{\mathcal{F}}}\nolimits\in \mathrm{FT}(C_6)$.
  • Figure 3: Example of $\mathop{\mathrm{rotatejoin}}\nolimits$ and $\mathop{\mathrm{rotatebreak}}\nolimits$.
  • Figure 4: The graph $G'$ next to $C_6+G'$.
  • Figure 5: A forest triple in $\mathrm{FT}(G)$ being broken into two parts.
  • ...and 7 more figures

Theorems & Definitions (68)

  • Definition
  • Definition
  • Remark
  • Definition
  • Definition
  • Example
  • Definition
  • Definition
  • Conjecture 2.2: STANLEY1993261
  • Definition
  • ...and 58 more