Partial-dual genus polynomial of graphs
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914016651116544 |
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| author | Cheng, Zhiyun |
| author_facet | Cheng, Zhiyun |
| contents | Recently, Chmutov introduced the partial duality of ribbon graphs, which can be regarded as a generalization of the classical Euler-Poincaré duality. The partial-dual genus polynomial $^\partial\varepsilon_G(z)$ is an enumeration of the partial duals of $G$ by Euler genus. For an intersection graph derived from a given chord diagram, the partial-dual genus polynomial can be defined by considering the ribbon graph associated to the chord diagram. In this paper, we provide a combinatorial approach to the partial-dual genus polynomial in terms of intersection graphs without referring to chord diagrams. After extending the definition of the partial-dual genus polynomial from intersection graphs to all graphs, we prove that it satisfies the four-term relation of graphs. This provides an answer to a problem proposed by Chmutov. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_18950 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Partial-dual genus polynomial of graphs Cheng, Zhiyun Combinatorics 05C10, 05C31 Recently, Chmutov introduced the partial duality of ribbon graphs, which can be regarded as a generalization of the classical Euler-Poincaré duality. The partial-dual genus polynomial $^\partial\varepsilon_G(z)$ is an enumeration of the partial duals of $G$ by Euler genus. For an intersection graph derived from a given chord diagram, the partial-dual genus polynomial can be defined by considering the ribbon graph associated to the chord diagram. In this paper, we provide a combinatorial approach to the partial-dual genus polynomial in terms of intersection graphs without referring to chord diagrams. After extending the definition of the partial-dual genus polynomial from intersection graphs to all graphs, we prove that it satisfies the four-term relation of graphs. This provides an answer to a problem proposed by Chmutov. |
| title | Partial-dual genus polynomial of graphs |
| topic | Combinatorics 05C10, 05C31 |
| url | https://arxiv.org/abs/2502.18950 |