On the $e$-positivity of trees and spiders

TL;DR

This paper proves that trees with a vertex of degree at least six are not $e$-positive in their chromatic symmetric functions, advancing a conjecture and providing conditions to identify non-$e$-positivity in spiders and related graphs.

Contribution

It establishes non-$e$-positivity for trees with high-degree vertices and offers new criteria for spiders, extending understanding of chromatic symmetric functions.

Findings

Trees with degree ≥6 are not $e$-positive.

Conditions to identify non-$e$-positivity in spiders.

Method to calculate coefficients in elementary symmetric expansion.

Abstract

We prove that for any tree with a vertex of degree at least six, its chromatic symmetric function is not $e$-positive, that is, it cannot be written as a nonnegative linear combination of elementary symmetric functions. This makes significant progress towards a recent conjecture of Dahlberg, She, and van Willigenburg, who conjectured the result for all trees with a vertex of degree at least four. We also provide a series of conditions that can identify when the chromatic symmetric function of a spider, a tree consisting of multiple paths identified at an end, is not $e$-positive. These conditions also generalize to trees and graphs with cut vertices. Finally, by applying a result of Orellana and Scott, we provide a method to inductively calculate certain coefficients in the elementary symmetric function expansion of the chromatic symmetric function of a spider, leading to further $e$-positivity conditions for spiders.