A Counterexample to a Conjecture on Schur Positivity of Chromatic Symmetric Functions of trees

Emmanuella Sandratra Rambeloson Emmanuella Sandratra Rambeloson,John Shareshian

doi:10.37236/9696

Emmanuella Sandratra Rambeloson Emmanuella Sandratra Rambeloson, John Shareshian

Open Access

https://doi.org/10.37236/9696

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Abstract

We show that no tree on twenty vertices with maximum degree ten has Schur positive chromatic symmetric function, thereby providing a counterexample to a conjecture of Dahlberg, She and van Willigenburg.

Highlights

O Theorem 39 for trees is posed in [1, Conjecture 42], which says that for every n 2, there is a tree T on n vertices, one of which has degree n 2
The chromatic symmetric function XG of G is the sum of all such monomials, XG(x) :=
Chromatic symmetric functions were introduced by Stanley in [5] and have drawn considerable attention

Summary

Introduction

O Theorem 39 for trees is posed in [1, Conjecture 42], which says that for every n 2, there is a tree T on n vertices, one of which has degree n 2. Such that the chromatic symmetric function of T is Schur positive. Given a (finite, loopless, simple) graph G = (V, E), a proper coloring of G is a function κ from V to the set P The chromatic symmetric function XG of G is the sum of all such monomials, XG(x) :=

Results

Conclusion