Chromatic symmetric functions and H-free graphs

Abstract

Using a graph and its colorings we can define a chromatic symmetric function. Stanley’s celebrated conjecture about the e-positivity of claw-free incomparability graphs has seen several related results, including one showing ( $$claw, P_4$$ )-free graphs are e-positive. 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.