We prove that the chromatic symmetric function of any n-vertex tree containing a vertex of degree d ≥ log 2 n + 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 ≥ log 2 n + 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 ⌈ n 2 ⌉ 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.

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We show that any Algebraic Branching Program (ABP) computing the polynomial ∑ n i=1 x n i has at least Ω(n 2 ) vertices. This improves upon the lower bound of Ω(n log n), which follows from the classical result of Baur and Strassen [Str73a, BS83], and extends the results in [Kum19], which showed a quadratic lower bound for homogeneous ABPs computing the same polynomial.Our proof relies on a notion of depth reduction which is reminiscent of similar statements in the context of matrix rigidity, and shows that any small enough ABP computing the polynomial ∑ n i=1 x n i can be depth reduced to essentially a homogeneous ABP of the same size which computes the polynomial ∑ n i=1 x n i + ε(x), for a structured "error polynomial" ε(x). To complete the proof, we then observe that the lower bound in [Kum19] is robust enough and continues to hold for all polynomials ∑ n i=1 x n i + ε(x), where ε(x) has the appropriate structure.

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