A quantitative way to e-positivity of trees

Abstract

In 2020, Dahlberg, She, and van Willigenburg conjectured that the chromatic symmetric function of any tree with maximum degree at least 4 is not e-positive. Zheng and Tom verified this conjecture for all trees with maximum degree at least 5 and spiders with maximum degree 4, and in their proofs the following necessary condition given by Wolfgang plays an important role: every connected graph having e-positive chromatic symmetric function must contain a connected partition of every type. In order to make further progress on this conjecture, we refine Wolfgang's result in a quantitative way. At first, we give an explicit formula for the e-coefficients of trees in terms of their connected partitions, by which e-positivity is equivalent to a series of inequalities for the numbers of connected partitions. Based on this formula, we present several necessary conditions on the numbers of connected partitions or acyclic orientations for trees to be e-positive. These necessary conditions turn out to be characterizations on the structure of e-positive trees, and as sample applications we prove the non-e-positivity of several classes of trees with maximum degree 3 or 4. We further make more discussions and calculations on trees with maximum degree 4 and having a connected partition of every type, which inspire us to come up with a list of open problems towards the final resolution of the above conjecture.