  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.

Combinatorics