Chromatic symmetric functions and H-free graphs

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.