Marked graphs and the chromatic symmetric function
Abstract
The main result of this paper is the introduction of marked graphs and the marked graph polynomials (-polynomial) associated with them. These polynomials can be defined via a deletion-contraction operation. These polynomials are a generalization of the -polynomial introduced by Noble and Welsh and a specialization of the -polynomial introduced by Ellis-Monaghan and Moffatt. In addition, we describe an important specialization of the -polynomial which we call the -polynomial. Furthermore, we give an efficient algorithm for computing the chromatic symmetric function of a graph in the \emph{star-basis} of symmetric functions. As an application of these tools, we prove that proper trees of diameter at most 5 can be reconstructed from its chromatic symmetric function.
Keywords
Cite
@article{arxiv.2202.11787,
title = {Marked graphs and the chromatic symmetric function},
author = {José Aliste-Prieto and Anna de Mier and Rosa Orellana and José Zamora},
journal= {arXiv preprint arXiv:2202.11787},
year = {2022}
}