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 ( c l a w , 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 = { c l a w , 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.