On the reconstruction of trees from their chromatic symmetric functions

TL;DR

This paper investigates the chromatic symmetric function of trees, providing a method to compute coefficients in the star basis, identifying key partitions, and offering an algorithm for reconstructing certain trees.

Contribution

It introduces a novel approach using the DNC algorithm to analyze the CSF in the star basis and presents a tree reconstruction algorithm for trees with diameter less than six.

Findings

Determined the smallest lexicographic partition in the CSF.

Derived a formula for the coefficient of this partition.

Developed an algorithm to reconstruct trees of diameter less than six.

Abstract

We study Stanley's chromatic symmetric function (CSF) for trees when expressed in the star basis. We use the deletion-near-contraction (DNC) algorithm to compute coefficients that occur in the CSF in the star basis. In particular, one of our main results determines the smallest partition in lexicographic order that occurs as an indexing partition in the CSF, and we also give a formula for its coefficient. In addition to describing properties of trees encoded in the coefficients of the star basis, we give an algorithm for reconstructing trees of diameter less than six.