Isometric embeddings of resonance graphs as finite distributive lattices

TL;DR

This paper investigates the isometric embedding of resonance graphs derived from plane bipartite graphs into hypercubes, establishing bounds on the embedding dimension and providing algorithms for efficient coding of perfect matchings.

Contribution

It characterizes when the resonance graph's isometric dimension equals the number of certain faces and offers an algorithm for binary coding of perfect matchings in these graphs.

Findings

The isometric dimension of the resonance graph is at least the number of specific faces without forbidden edges.

Conditions are provided for when the isometric dimension equals this face count.

An algorithm is developed to generate binary codes for perfect matchings, enabling isometric embedding without enumerating all matchings.

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.]