The Chromatic Symmetric Function of Almost Complete Graphs

The chromatic symmetric function of a graph is a symmetric function that generalizes the chromatic polynomial. Its investigation has largely been motivated by the existence of an open problem, the poset-chain conjecture, which is equivalent to the assertion that for certain graphs, the coefficients in the expansion of the chromatic symmetric function in terms of elementary symmetric functions, known as the e-coefficients of the graph, are non-negative. In this paper we study how the e-coefficients of a graph G are affected by taking the disjoint union of G and a complete graph, plus all edges between the vertices of G and the vertices of the complete graph. We call such a graph an almost complete graph. Our main result is a description of the chromatic symmetric function of almost complete graphs that can be used to study their e-coefficients.