An extension of Stanley's chromatic symmetric function to binary delta-matroids

Abstract

Stanley's symmetrized chromatic polynomial is a generalization of the ordinary chromatic polynomial to a graph invariant with values in a ring of polynomials in infinitely many variables. The ordinary chromatic polynomial is a specialization of Stanley's one.To each orientable embedded graph with a single vertex, a simple graph is associated, which is called the intersection graph of the embedded graph. As a result, we can define Stanley's symmetrized chromatic polynomial for any orientable embedded graph with a single vertex. Our goal is to extend Stanley's chromatic polynomial to embedded graphs with arbitrary number of vertices, and not necessarily orientable. In contrast to well-known extensions of, say, the Tutte polynomial from abstract to embedded graphs [4], our extension is based not on the structure of the underlying abstract graph and the additional information about the embedding. Instead, we consider the binary delta-matroid associated to an embedded graph and define the extended Stanley chromatic polynomial as an invariant of binary delta-matroids. We show that, similarly to Stanley's symmetrized chromatic polynomial of graphs, which satisfies 4-term relations for simple graphs, the polynomial that we introduce satisfies the 4-term relations for binary delta-matroids [7].For graphs, Stanley's chromatic function produces a knot invariant by means of the correspondence between simple graphs and knots. Analogously we may interpret the suggested extension as an invariant of links, using the correspondence between binary delta-matroids and links.