A decomposition structure of resonance graphs that are daisy cubes

TL;DR

This paper introduces a decomposition structure for resonance graphs of certain bipartite graphs, providing an algorithm to embed these graphs into hypercubes and comparing different binary codings.

Contribution

It presents a novel decomposition method for resonance graphs of peripherally 2-colorable bipartite graphs and an algorithm for proper labelling and hypercube embedding.

Findings

Resonance graphs of certain bipartite graphs are daisy cubes.

An algorithm for generating proper labelling of perfect matchings.

Comparison of binary codings inducing different graph structures.

Abstract

It has recently been shown in [\emph{Discrete Appl. Math.} {\bf 366} (2025) 75--85] that the resonance graph of a plane elementary bipartite graph $G$ is a daisy cube if and only if $G$ is peripherally 2-colorable. Let $G$ be a peripherally 2-colorable graph and $R(G)$ be its resonance graph. We provide a decomposition structure of $R(G)$ with respect to an arbitrary finite face of $G$ together with a proper labelling for the vertex set of $R(G)$. An algorithm is obtained to generate a proper labelling for all perfect matchings of $G$ which induces an isometric embedding of $R(G)$ as a daisy cube into an $n$-dimensional hypercube, where $n$ is the isometric dimension of $R(G)$. Moreover, the algorithm can be applied to generate such a proper labelling for all perfect matchings of any plane weakly elementary bipartite graph whose each elementary component with more than two vertices is peripherally 2-colorable. We also compare two binary codings for all perfect matchings of $G$ which induces distinct structures on $R(G)$: one as a daisy cube and the other as a finite distributive, respectively.