$e$-Positive caterpillars have maximum degree at most three

Abstract

Dahlberg, She, and van Willigenburg conjectured that every $e$-positive tree has maximum degree at most three. We prove this conjecture for caterpillars. Our proof introduces a specialization--cone method: we evaluate chromatic symmetric functions under finite-support specializations that are nonnegative on the elementary symmetric functions, express the evaluations through transfer matrices, and certify negativity using finitely generated cones. Two specializations and three cones handle all caterpillars remaining after a connected-partition reduction, with no bound on the number or spacing of their trivalent vertices. The required cone inclusions and strict sign inequalities are certified by exact rational arithmetic. We also exhibit two infinite families of non-spider caterpillars of maximum degree four that admit connected partitions of every type, showing that Wolfgang's criterion alone leaves cases requiring further obstructions. Our result also reduces Li's criterion on $e$-positivity for caterpillars of maximum degree three to an explicit condition on the positions of trivalent vertices.