Chromatic symmetric functions and H-free graphs

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,P4)-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.