Distinguishability and linear independence for $H$-chromatic symmetric functions
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866918197096087552 |
|---|---|
| author | Lin, Shao Yuan Pierson, Laura |
| author_facet | Lin, Shao Yuan Pierson, Laura |
| contents | We study the $H$-chromatic symmetric functions $X_G^H$ (introduced in (arXiv:2011.06063) as a generalization of the chromatic symmetric function (CSF) $X_G$), which track homomorphisms from the graph $G$ to the graph $H$. We focus first on the case of self-chromatic symmetric functions (self-CSFs) $X_G^G$, making some progress toward a conjecture from (arXiv:2011.06063) that the self-CSF, like the normal CSF, is always different for different trees. In particular, we show that the self-CSF distinguishes trees from non-trees with just one exception, we check using Sage that it distinguishes all trees on up to 12 vertices, and we show that it determines the number of legs of a spider and the degree sequence of a caterpillar given its spine length. We also show that the self-CSF detects the number of connected components of a forest, again with just one exception. Then we prove some results about the power sum expansions for $H$-CSFs when $H$ is a complete bipartite graph, in particular proving that the conjecture from (arXiv:2011.06063) about $p$-monotonicity of $ω(X_G^H)$ for $H$ a star holds as long as $H$ is sufficiently large compared to $G$. We also show that the self-CSFs of complete multipartite graphs form a basis for the ring $Λ$ of symmetric functions, and we give some construction of bases for the vector space $Λ^n$ of degree $n$ symmetric functions using $H$-CSFs $X_G^H$ where $H$ is a fixed graph that is not a complete graph, answering a question from (arXiv:2011.06063) about whether such bases exist. However, we show that there generally do not exist such bases with $G$ fixed, even with loops, answering another question from (arXiv:2011.06063). We also define the $H$-chromatic polynomial as an analogue of the chromatic polynomial, and ask when it is the same for different graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_08665 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Distinguishability and linear independence for $H$-chromatic symmetric functions Lin, Shao Yuan Pierson, Laura Combinatorics 05E05, 05C25, 05C31, 05C50, 05C60 We study the $H$-chromatic symmetric functions $X_G^H$ (introduced in (arXiv:2011.06063) as a generalization of the chromatic symmetric function (CSF) $X_G$), which track homomorphisms from the graph $G$ to the graph $H$. We focus first on the case of self-chromatic symmetric functions (self-CSFs) $X_G^G$, making some progress toward a conjecture from (arXiv:2011.06063) that the self-CSF, like the normal CSF, is always different for different trees. In particular, we show that the self-CSF distinguishes trees from non-trees with just one exception, we check using Sage that it distinguishes all trees on up to 12 vertices, and we show that it determines the number of legs of a spider and the degree sequence of a caterpillar given its spine length. We also show that the self-CSF detects the number of connected components of a forest, again with just one exception. Then we prove some results about the power sum expansions for $H$-CSFs when $H$ is a complete bipartite graph, in particular proving that the conjecture from (arXiv:2011.06063) about $p$-monotonicity of $ω(X_G^H)$ for $H$ a star holds as long as $H$ is sufficiently large compared to $G$. We also show that the self-CSFs of complete multipartite graphs form a basis for the ring $Λ$ of symmetric functions, and we give some construction of bases for the vector space $Λ^n$ of degree $n$ symmetric functions using $H$-CSFs $X_G^H$ where $H$ is a fixed graph that is not a complete graph, answering a question from (arXiv:2011.06063) about whether such bases exist. However, we show that there generally do not exist such bases with $G$ fixed, even with loops, answering another question from (arXiv:2011.06063). We also define the $H$-chromatic polynomial as an analogue of the chromatic polynomial, and ask when it is the same for different graphs. |
| title | Distinguishability and linear independence for $H$-chromatic symmetric functions |
| topic | Combinatorics 05E05, 05C25, 05C31, 05C50, 05C60 |
| url | https://arxiv.org/abs/2511.08665 |