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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2405.18862 |
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| _version_ | 1866912981780004864 |
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| author | Brezovnik, Simon Che, Zhongyuan Tratnik, Niko Pleteršek, Petra Žigert |
| author_facet | Brezovnik, Simon Che, Zhongyuan Tratnik, Niko Pleteršek, Petra Žigert |
| contents | Let $G$ be a plane elementary bipartite graph whose infinite face is forcing. We provide a bijection between the set of maximal hypercubes of its resonance graph and the set of maximal resonant sets of $G$, which generalizes a main result in [MATCH Commun. Math. Comput. Chem. 68 (2012) 65-77], where $G$ was only considered as an elementary benzenoid graph without nice coronenes. For a special case when $G$ is a peripherally 2-colorable graph, it follows that there is a bijection between the set of maximal hypercubes of its resonance graph and the set of maximal independent sets of a tree that is the inner dual of $G$. We then show that the resonance graph of a plane bipartite graph $G$ is a daisy cube if and only if it is the simplex graph of the complement of a forest. Finally, we characterize trees with at most 5 maximal independent sets to determine daisy cubes that are simplex graphs of the complements of trees and having at most five maximal vertices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_18862 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Resonance graphs that are daisy cubes: from hypercubes to independent sets via resonant sets Brezovnik, Simon Che, Zhongyuan Tratnik, Niko Pleteršek, Petra Žigert Combinatorics 05C70 05C69 05C10 05C05 05C92 Let $G$ be a plane elementary bipartite graph whose infinite face is forcing. We provide a bijection between the set of maximal hypercubes of its resonance graph and the set of maximal resonant sets of $G$, which generalizes a main result in [MATCH Commun. Math. Comput. Chem. 68 (2012) 65-77], where $G$ was only considered as an elementary benzenoid graph without nice coronenes. For a special case when $G$ is a peripherally 2-colorable graph, it follows that there is a bijection between the set of maximal hypercubes of its resonance graph and the set of maximal independent sets of a tree that is the inner dual of $G$. We then show that the resonance graph of a plane bipartite graph $G$ is a daisy cube if and only if it is the simplex graph of the complement of a forest. Finally, we characterize trees with at most 5 maximal independent sets to determine daisy cubes that are simplex graphs of the complements of trees and having at most five maximal vertices. |
| title | Resonance graphs that are daisy cubes: from hypercubes to independent sets via resonant sets |
| topic | Combinatorics 05C70 05C69 05C10 05C05 05C92 |
| url | https://arxiv.org/abs/2405.18862 |