All cycle-chords are $e$-positive

TL;DR

This paper proves that cycle-chord graphs are $e$-positive using a new, simpler method, offers a combinatorial interpretation of the coefficients, and conjectures $e$-positivity for theta graphs.

Contribution

It introduces a simplified composition method to establish $e$-positivity of cycle-chord graphs and provides a combinatorial interpretation of the coefficients.

Findings

Cycle-chord graphs are $e$-positive.

A combinatorial interpretation of $e$-coefficients is provided.

Conjecture that theta graphs are $e$-positive.

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.