Classes of graphs with e-positive chromatic symmetric function

TL;DR

This paper investigates classes of graphs with e-positive chromatic symmetric functions, proving e-positivity for certain unit interval graphs and proposing a conjecture linking strong e-positivity to (claw, net)-free graphs.

Contribution

It proves e-positivity for unit interval graphs with complements also being unit interval graphs and introduces the concept of strong e-positivity with a related conjecture.

Findings

Proved e-positivity for a new class of unit interval graphs.

Introduced the concept of strongly e-positive graphs.

Formulated a conjecture relating strong e-positivity to (claw, net)-free graphs.

Abstract

In the mid-1990s, Stanley and Stembridge conjectured that the chromatic symmetric functions of claw-free co-comparability (also called incomparability) graphs were e-positive. The quest for the proof of this conjecture has led to an examination of other, related graph classes. In 2013 Guay-Paquet proved that if unit interval graphs are e-positive, that implies claw-free incomparability graphs are as well. Inspired by this approach, we consider a related case and prove that unit interval graphs whose complement is also a unit interval graph are e-positive. We introduce the concept of strongly $e$-positive to denote a graph whose induced subgraphs are all e-positive, and conjecture that a graph is strongly e-positive if and only if it is (claw, net)-free.