# Resolving Stanley's $e$-positivity of claw-contractible-free graphs

## TL;DR

This paper identifies infinite families of graphs that are not contractible to the claw and have chromatic symmetric functions that are not $e$-positive, challenging previous assumptions about the relationship between claw-contractibility and $e$-positivity.

## Contribution

It provides the first explicit examples of non-contractible graphs with non-$e$-positive chromatic symmetric functions, including claw-free graphs, clarifying the independence of $e$-positivity from claw structures.

## Key findings

- Identified infinite families of non-contractible graphs with non-$e$-positive functions.
- Established that $e$-positivity is not dependent on the presence of an induced claw.
- Provided examples of claw-free graphs with non-$e$-positive chromatic symmetric functions.

## Abstract

In Stanley's seminal 1995 paper on the chromatic symmetric function, he stated that there was no known graph that was not contractible to the claw and whose chromatic symmetric function was not $e$-positive, namely, not a positive linear combination of elementary symmetric functions. We resolve this by giving infinite families of graphs that are not contractible to the claw and whose chromatic symmetric functions are not $e$-positive. Moreover, one such family is additionally claw-free, thus establishing that the $e$-positivity of chromatic symmetric functions is in general not dependent on the existence of an induced claw or of a contraction to a claw.