Chromatic Posets

2019/09/26 by Samantha Dahlberg, Dahlberg, Samantha, Adrian She +3
Computer Science · Mathematics · #05C15 #05C25 #05E05 (Primary) #06A06 (Secondary) #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems #Graph theory and applications

paper · pdf · doi:10.48550/arxiv.1909.12394

openalex publication_date 2019/09/26 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28

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.

Related

• Planar graphs and Stanley's Chromatic Functions
• The e-positivity and Schur positivity of the chromatic symmetric functions of some trees
• Schur and e-positivity of trees and cut vertices
• Hook coefficients of chromatic functions
• Non-Schur-positivity of chromatic symmetric functions