Counting induced subgraphs with the Kromatic symmetric function
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866915161987612672 |
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| author | Pierson, Laura |
| author_facet | Pierson, Laura |
| contents | The chromatic symmetric function $X_G$ is a sum of monomials corresponding to proper vertex colorings of a graph $G$. Crew, Pechenik, and Spirkl (2023) recently introduced a $K$-theoretic analogue $\overline{X}_G$ called the Kromatic symmetric function, where each vertex is instead assigned a nonempty set of colors such that adjacent vertices have nonoverlapping color sets. $X_G$ does not distinguish all graphs, but a longstanding open question is whether it distinguishes all trees. We conjecture that $\overline{X}_G$ does distinguish all graphs. As evidence towards this conjecture, we show that $\overline{X}_G$ determines the number of copies in $G$ of certain induced subgraphs on 4 and 5 vertices as well as the number of induced subgraphs isomorphic to each graph consisting of a star plus some number of isolated vertices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_15929 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Counting induced subgraphs with the Kromatic symmetric function Pierson, Laura Combinatorics The chromatic symmetric function $X_G$ is a sum of monomials corresponding to proper vertex colorings of a graph $G$. Crew, Pechenik, and Spirkl (2023) recently introduced a $K$-theoretic analogue $\overline{X}_G$ called the Kromatic symmetric function, where each vertex is instead assigned a nonempty set of colors such that adjacent vertices have nonoverlapping color sets. $X_G$ does not distinguish all graphs, but a longstanding open question is whether it distinguishes all trees. We conjecture that $\overline{X}_G$ does distinguish all graphs. As evidence towards this conjecture, we show that $\overline{X}_G$ determines the number of copies in $G$ of certain induced subgraphs on 4 and 5 vertices as well as the number of induced subgraphs isomorphic to each graph consisting of a star plus some number of isolated vertices. |
| title | Counting induced subgraphs with the Kromatic symmetric function |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2403.15929 |