# Schur and $e$-positivity of trees and cut vertices

## TL;DR

This paper investigates conditions under which the chromatic symmetric function of trees and graphs is not $e$-positive or Schur-positive, revealing that high-degree vertices and certain structural features prevent positivity.

## Contribution

It establishes new non-positivity results for chromatic symmetric functions based on vertex degrees and graph connectivity, extending understanding of $e$-positivity and Schur-positivity.

## Key findings

- High-degree vertices in trees imply non-$e$-positivity.
- Presence of cut vertices with many components leads to non-$e$-positivity.
- Graphs without perfect or almost perfect matchings are not $e$-positive.

## Abstract

We prove that the chromatic symmetric function of any $n$-vertex tree containing a vertex of degree $d\geq \log _2n +1$ is not $e$-positive, that is, not a positive linear combination of elementary symmetric functions. Generalizing this, we also prove that the chromatic symmetric function of any $n$-vertex connected graph containing a cut vertex whose deletion disconnects the graph into $d\geq\log _2n +1$ connected components is not $e$-positive. Furthermore we prove that any $n$-vertex bipartite graph, including all trees, containing a vertex of degree greater than $\lceil \frac{n}{2}\rceil$ is not Schur-positive, namely not a positive linear combination of Schur functions. In complete generality, we prove that if an $n$-vertex connected graph has no perfect matching (if $n$ is even) or no almost perfect matching (if $n$ is odd), then it is not $e$-positive. We hence deduce that many graphs containing the claw are not $e$-positive.