Abstract
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.
Highlights
In order to describe our results, let us first recall the necessary combinatorics and algebra.We say a partition λ = (λ1, . . . , λ (λ)) of N, denoted by λ N, is a list of positive integers whose parts λi satisfy λ1λ (λ) > 0 and (λ) i=1 λi = N.If we have j parts equal to i we often denote this by ij
The electronic journal of combinatorics 27(1) (2020), #P1.2 In Section 2 we review the necessary notions before reducing the graphs we need to study to spiders in Subsection 2.1
We say that a connected graph G contains a cut vertex if there exists a vertex v ∈ VG such that the deletion of v and its incident edges yields the electronic journal of combinatorics 27(1) (2020), #P1.2 a graph G with more than one connected component
Summary
In order to describe our results, let us first recall the necessary combinatorics and algebra. If T has no perfect matching (if n is even) or no almost perfect matching (if n is odd), T is not e-positive It is a well-known result that if a connected graph with an even number of vertices is claw-free, it has a perfect matching. This immediately yields the following corollary to Theorem 7. The n-vertex star Sn for n 4 is not e-positive since it is missing a connected partition of type (n − 2, 2) by Example 1. It is not e-positive since it is missing for n even and. That the converse of Theorem 6 and of Theorem 7 is false since the spider S(4, 1, 1) is not e-positive by Example 5 and yet has a connected partition of every type
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