Schur and \(e\)-positivity of trees and cut vertices (Q2288162)

Summary: We prove that the chromatic symmetric function of any \(n\)-vertex tree containing a vertex of degree \(d\geqslant \log_2n +1\) is not \(e\)-positive, that is, not a positive linear combination of elementary symmetric functions. Generalizing this, we also prove that the chromatic symmetric function of any \(n\)-vertex connected graph containing a cut vertex whose deletion disconnects the graph into \(d\geqslant\log_2n +1\) connected components is not \(e\)-positive. Furthermore we prove that any \(n\)-vertex bipartite graph, including all trees, containing a vertex of degree greater than \(\lceil \frac{n}{2}\rceil\) is not Schur-positive, namely not a positive linear combination of Schur functions. In complete generality, we prove that if an \(n\)-vertex connected graph has no perfect matching (if \(n\) is even) or no almost perfect matching (if \(n\) is odd), then it is not \(e\)-positive. We hence deduce that many graphs containing the claw are not \(e\)-positive.