Chromatic symmetric functions and \(H\)-free graphs. (English) Zbl 1416.05105
Summary: 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 (\(\text{claw}, P_4\))-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 \(H = \{\text{claw}, F\}\) and \(H=\text{claw}, F, \text{co-}F\), where \(F\) is a four-vertex graph. We settle the question for all cases except \(H=\text{claw}, \text{co-diamond}\), and we provide some partial results in that case.
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