The $e$-positivity and Schur positivity of the chromatic symmetric functions of some trees

TL;DR

This paper studies the positivity properties of chromatic symmetric functions for certain trees, classifies positivity for broom and double broom graphs, and proposes conjectures for broader classes.

Contribution

It provides a classification of $e$-positivity and Schur positivity for specific spider graphs, introducing new methods and conjectures in the study of chromatic symmetric functions.

Findings

Classified positivity for broom graphs

Classified most double broom graphs

Proposed conjectures on positivity of trees

Abstract

We investigate the $e$-positivity and Schur positivity of the chromatic symmetric functions of some spider graphs with three legs. We obtain the positivity classification of all broom graphs and that of most double broom graphs. The methods involve extracting particular $e$-coefficients of the chromatic symmetric function of these graphs with the aid of Orellana and Scott's triple-deletion property, and using the combinatorial formula of Schur coefficients by examining certain special rim hook tabloids. We also propose some conjectures on the $e$-positivity and Schur positivity of trees.