  In 1995 Stanley introduced the chromatic symmetric function $X_G$ 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 $X_G$ and
$X_H$. We find a biconditional condition 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 trees form 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 form a chain.

Combinatorics