Motivated by the notion of nice graphs, we introduce the concept of strongly nice property, which can be used to study the Schur positivity of symmetric functions. We show that a graph and all its induced subgraphs are strongly nice if and only if it is claw-free, which strengthens a result of Stanley and provides further evidence for the well-known conjecture on the Schur positivity of claw-free graphs. As another application, we solve Wang and Wang's conjecture on the non-Schur positivity of squid graphs Sq(2n-1;1ⁿ) for n ≥ 3 by proving that these graphs are not strongly nice.
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Li et al. (2024) studied this question.
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