Strongly nice property and Schur positivity of graphs

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 $Sq(2n-1;1^n)$ for $n \ge 3$ by proving that these graphs are not strongly nice.