Chromatic Posets
| Autor: | Samantha Dahlberg, Adrian She, Stephanie van Willigenburg |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
Mathematics::Combinatorics
Computational Theory and Mathematics 010201 computation theory & mathematics 010102 general mathematics FOS: Mathematics Discrete Mathematics and Combinatorics Mathematics - Combinatorics 0102 computer and information sciences Combinatorics (math.CO) 0101 mathematics 01 natural sciences 05E05 (Primary) 05C15 05C25 06A06 (Secondary) Theoretical Computer Science |
| DOI: | 10.48550/arxiv.1909.12394 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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