Isometric embeddings of resonance graphs as finite distributive lattices

Abstract

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.]

Figures (3)

• Figure 1: An example for implementing Algorithm \ref{['A:FDL']}.
• Figure 2: Coronene.
• Figure 3: A plane weakly elementary bipartite graph $G$ with two elementary components $E_1$ and $E_2$, and a binary coding $[\phi]=(\phi^{[E_1]}, \phi^{[E_2]})$ of length $4$ on $\mathcal{M}(G)$.

Theorems & Definitions (19)

• Definition 2.1
• Lemma 2.2
• Definition 2.3
• Theorem 2.4
• Corollary 2.5
• Remark 2.1
• Lemma 2.6
• Theorem 3.1
• Corollary 3.2
• Theorem 3.4