Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A decomposition structure of resonance graphs that are daisy cubes

Zhongyuan Che, Zhibo Chen

TL;DR

This work addresses the structural understanding of resonance graphs $R(G)$ for plane bipartite graphs, proving that $R(G)$ is a daisy cube precisely when $G$ is peripherally 2-colorable. It introduces a decomposition framework with respect to a finite face $s$ and a constructive proper labeling that yields an isometric embedding of $R(G)$ into an $n$-dimensional hypercube, with an algorithm to generate the labeling for all perfect matchings. The results extend to plane weakly elementary bipartite graphs whose nontrivial elementary components are peripherally 2-colorable and provide two distinct binary codings: one embedding $R(G)$ as a finite distributive lattice and another as a daisy cube, revealing differing structural insights. Collectively, these findings offer a practical, constructive approach to understanding resonance graphs, enabling isometric embeddings and enabling comparisons between daisy-cube and distributive-lattice representations of $R(G)$, with potential implications for related combinatorial chemistry models and graph-theoretic analyses.

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.

A decomposition structure of resonance graphs that are daisy cubes

TL;DR

This work addresses the structural understanding of resonance graphs for plane bipartite graphs, proving that is a daisy cube precisely when is peripherally 2-colorable. It introduces a decomposition framework with respect to a finite face and a constructive proper labeling that yields an isometric embedding of into an -dimensional hypercube, with an algorithm to generate the labeling for all perfect matchings. The results extend to plane weakly elementary bipartite graphs whose nontrivial elementary components are peripherally 2-colorable and provide two distinct binary codings: one embedding as a finite distributive lattice and another as a daisy cube, revealing differing structural insights. Collectively, these findings offer a practical, constructive approach to understanding resonance graphs, enabling isometric embeddings and enabling comparisons between daisy-cube and distributive-lattice representations of , with potential implications for related combinatorial chemistry models and graph-theoretic analyses.

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 is a daisy cube if and only if is peripherally 2-colorable. Let be a peripherally 2-colorable graph and be its resonance graph. We provide a decomposition structure of with respect to an arbitrary finite face of together with a proper labelling for the vertex set of . An algorithm is obtained to generate a proper labelling for all perfect matchings of which induces an isometric embedding of as a daisy cube into an -dimensional hypercube, where is the isometric dimension of . 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 which induces distinct structures on : one as a daisy cube and the other as a finite distributive, respectively.
Paper Structure (8 sections, 8 theorems, 1 equation, 1 figure, 1 algorithm)

This paper contains 8 sections, 8 theorems, 1 equation, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Convex expansions and median graphs
  4. Daisy cubes
  5. Decompositions of resonance graphs
  6. Subsets of perfect matchings of peripherally 2-colorable graphs
  7. Main Results
  8. Two distinct binary codings

Key Result

Theorem 2.1

HIK11 Let $xy$ be an edge of a connected bipartite graph $G$. Then $G$ is a median graph if and only if the following three conditions are satisfied: (i) $F_{xy}$ is a matching defining an isomorphism between $\langle U_{xy} \rangle$ and $\langle U_{yx} \rangle$, (ii) $\langle U_{xy} \rangle$ is con

Figures (1)

  • Figure 1: An example for implementing Algorithm \ref{['A:DaisyCube']}.

Theorems & Definitions (10)

  • Theorem 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.5
  • Lemma 3.1
  • Theorem 3.2
  • Corollary 3.3
  • Corollary 3.4
  • Corollary 3.6
  • Corollary 4.1