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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2410.07411 |
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| _version_ | 1866909344129351680 |
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| author | Che, Zhongyuan |
| author_facet | Che, Zhongyuan |
| contents | Let $G$ be a plane bipartite graph and $\mathcal{M}(G)$ be the set of all perfect matchings of $G$. The resonance graph $R(G)$ is a graph whose vertex set is $\mathcal{M}(G)$, and two perfect matchings are adjacent in $R(G)$ if their symmetric difference is a cycle forming the periphery of a finite face of $G$. It is known that any connected resonance graph can be isometrically embedded as a finite distributive lattice into hypercubes. The isometric dimension of a connected $R(G)$, denoted by $\mathrm{idim}(R(G))$, is the smallest dimension of a hypercube that $R(G)$ can be isometrically embedded into. Let $d$ be the number of finite faces of $G$ such that there are no forbidden edges on their peripheries. We show that any connected $R(G)$ has $\mathrm{idim}(R(G)) \ge d$ and provide characterizations on when the equality holds. Moreover, if a connected $R(G)$ has $\mathrm{idim}(R(G)) = d$, then we design an algorithm to generate a binary coding of length $d$ for all perfect matchings of $G$ which induces an isometric embedding of $R(G)$ as a finite distributive lattice into a $d$-dimensional hypercube without generating $\mathcal{M}(G)$. Our results provide answers for the fundamental cases of both open questions raised in [\textit{SIAM J. Discrete Math.} {\bf 22} (2008) 971--984.] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_07411 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Isometric embeddings of resonance graphs as finite distributive lattices Che, Zhongyuan Combinatorics Let $G$ be a plane bipartite graph and $\mathcal{M}(G)$ be the set of all perfect matchings of $G$. The resonance graph $R(G)$ is a graph whose vertex set is $\mathcal{M}(G)$, and two perfect matchings are adjacent in $R(G)$ if their symmetric difference is a cycle forming the periphery of a finite face of $G$. It is known that any connected resonance graph can be isometrically embedded as a finite distributive lattice into hypercubes. The isometric dimension of a connected $R(G)$, denoted by $\mathrm{idim}(R(G))$, is the smallest dimension of a hypercube that $R(G)$ can be isometrically embedded into. Let $d$ be the number of finite faces of $G$ such that there are no forbidden edges on their peripheries. We show that any connected $R(G)$ has $\mathrm{idim}(R(G)) \ge d$ and provide characterizations on when the equality holds. Moreover, if a connected $R(G)$ has $\mathrm{idim}(R(G)) = d$, then we design an algorithm to generate a binary coding of length $d$ for all perfect matchings of $G$ which induces an isometric embedding of $R(G)$ as a finite distributive lattice into a $d$-dimensional hypercube without generating $\mathcal{M}(G)$. Our results provide answers for the fundamental cases of both open questions raised in [\textit{SIAM J. Discrete Math.} {\bf 22} (2008) 971--984.] |
| title | Isometric embeddings of resonance graphs as finite distributive lattices |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2410.07411 |