Chromatic symmetric functions and H-free graphs

TL;DR

This paper investigates the positivity of chromatic symmetric functions in H-free graphs, extending known conjectures from claw-free graphs to broader classes and providing new results and partial solutions.

Contribution

It generalizes the positivity conjecture to H-free graphs with specific forbidden subgraphs, solving most cases and offering partial results for the remaining one.

Findings

Confirmed e-positivity for most H-free graph classes

Provided partial results for the case H={claw, co-diamond}

Extended the scope of chromatic symmetric function positivity conjectures

Abstract

Two celebrated conjectures in chromatic symmetric function theory concern the positivity chromatics symmetric functions of claw-free graphs. Here we extend the claw-free idea to general graphs and consider the e-positivity question for H-free graphs where H = {claw, F} and H={claw, F, co-F}, where F is a four-vertex graph. We settle the question for all cases except H={claw, co-diamond}, and we provide some partial results in that case.