The chromatic symmetric function of graphs glued at a single vertex
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915213024952320 |
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| author | Tom, Foster Vailaya, Aarush |
| author_facet | Tom, Foster Vailaya, Aarush |
| contents | We describe how the chromatic symmetric function of two graphs glued at a single vertex can be expressed as a matrix multiplication using certain information of the two individual graphs. We then prove new $e$-positivity results by using a connection between forest triples, defined by the first author, and Hikita's probabilities associated to standard Young tableaux. Specifically, we prove that gluing a sequence of unit interval graphs and cycles results in an $e$-positive graph. We also prove $e$-positivity for a graph obtained by gluing the first and last vertices of such a sequence. This generalizes $e$-positivity of cycle-chord graphs and supports Ellzey's conjectured $e$-positivity for proper circular arc digraphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_19344 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The chromatic symmetric function of graphs glued at a single vertex Tom, Foster Vailaya, Aarush Combinatorics Primary 05E05, Secondary 05E10, 05C15 We describe how the chromatic symmetric function of two graphs glued at a single vertex can be expressed as a matrix multiplication using certain information of the two individual graphs. We then prove new $e$-positivity results by using a connection between forest triples, defined by the first author, and Hikita's probabilities associated to standard Young tableaux. Specifically, we prove that gluing a sequence of unit interval graphs and cycles results in an $e$-positive graph. We also prove $e$-positivity for a graph obtained by gluing the first and last vertices of such a sequence. This generalizes $e$-positivity of cycle-chord graphs and supports Ellzey's conjectured $e$-positivity for proper circular arc digraphs. |
| title | The chromatic symmetric function of graphs glued at a single vertex |
| topic | Combinatorics Primary 05E05, Secondary 05E10, 05C15 |
| url | https://arxiv.org/abs/2503.19344 |