Chromatic Posets

TL;DR

This paper explores the relationships between chromatic symmetric functions of graphs, introducing posets based on $e$-positivity and Schur-positivity, revealing structural properties and new paradigms in graph invariants.

Contribution

It defines and studies new posets on chromatic symmetric functions based on positivity relations, providing a novel framework for understanding graph invariants.

Findings

Trees form an independent set and are maximal elements.

Stars are independent elements in the posets.

Lollipop graphs form a chain within the poset.

Abstract

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.