Chromatic posets

Abstract

In 1995 Stanley introduced the chromatic symmetric function XG of a graph G, whose e-positivity and Schur-positivity has been of large interest. In this paper we study the relative e-positivity and Schur-positivity between connected graphs on n vertices. We define and investigate two families of posets on distinct chromatic symmetric functions. The relations depend on the e-positivity or Schur-positivity of a weighed subtraction between XG and XH. We find a biconditional criterion between e-positivity or Schur-positivity and the relation to the complete graph. This gives a new paradigm for e-positivity and for Schur-positivity. We show many other interesting properties of these posets including that the family of trees forms an independent set and are maximal elements. Additionally, we find that stars are independent elements, the independence number increases as we increase in the poset and that the family of lollipop graphs forms a chain.