MARKED GRAPHS AND THE CHROMATIC SYMMETRIC FUNCTION

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 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 present an efficient algorithm for computing the chromatic symmetric function of a graph in the star basis of symmetric functions. As an application of these tools, we prove that proper trees of diameter at most 5 are reconstructible from its chromatic symmetric function.