On the e-positivity of trees and spiders

Abstract

We prove that the chromatic symmetric function of any tree with a vertex of degree at least six 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 the chromatic symmetric functions of 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, which is a tree consisting of multiple paths all adjacent at a leaf to a center vertex, is not e-positive. These conditions generalize to trees and graphs with cut vertices as well. Finally, by using a result of Orellana and Scott, we give 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.