Marked graphs and the chromatic symmetric function

TL;DR

This paper introduces marked graphs and associated polynomials, providing new tools for analyzing the chromatic symmetric function and demonstrating tree reconstructibility from this function.

Contribution

It defines marked graphs and the $M$-polynomial, generalizes existing polynomials, and offers an efficient algorithm for computing the chromatic symmetric function in the star-basis.

Findings

Proper trees of diameter at most 5 can be reconstructed from their chromatic symmetric function.

Introduces the $M$-polynomial as a generalization of existing graph polynomials.

Provides an efficient algorithm for computing the chromatic symmetric function.

Abstract

The main result of this paper is the introduction of marked graphs and the marked graph polynomials ($M$-polynomial) associated with them. These polynomials can be defined via a deletion-contraction operation. These polynomials are a generalization of the $W$-polynomial introduced by Noble and Welsh and a specialization of the $\mathbf{V}$-polynomial introduced by Ellis-Monaghan and Moffatt. In addition, we describe an important specialization of the $M$-polynomial which we call the $D$-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.