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Isomorphic daisy cubes based on their $τ$-graphs

Zhongyuan Che, Niko Tratnik, Petra Žigert Pleteršek

Abstract

We prove that if $A$ and $B$ are daisy cubes whose $τ$-graphs are forests, then $A$ and $B$ are isomorphic if and only if their $τ$-graphs are isomorphic. The result is applied to show that a daisy cube with at least one edge is the resonance graph of a plane bipartite graph $G$ if and only if its $τ$-graph is a forest which is isomorphic to the inner dual of the subgraph of $G$ obtained by removing all forbidden edges. As a consequence, some well known properties of Fibonacci cubes and Lucas cubes are provided as examples with different proofs.

Isomorphic daisy cubes based on their $τ$-graphs

Abstract

We prove that if and are daisy cubes whose -graphs are forests, then and are isomorphic if and only if their -graphs are isomorphic. The result is applied to show that a daisy cube with at least one edge is the resonance graph of a plane bipartite graph if and only if its -graph is a forest which is isomorphic to the inner dual of the subgraph of obtained by removing all forbidden edges. As a consequence, some well known properties of Fibonacci cubes and Lucas cubes are provided as examples with different proofs.

Paper Structure

This paper contains 4 sections, 8 theorems, 1 equation, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Isomorphic daisy cubes based on their $\tau$-graphs
  4. Daisy cubes that are resonance graphs

Key Result

Lemma 3.1

Let $H$ be a daisy cube with $n$$\Theta$-classes for some positive integer $n$. Then $H=Q_n$ if and only if $H^{\tau}=K^C_n$.

Figures (4)

  • Figure 1: Partial cubes $C_6$, $\Lambda_3$, and $Q_3^{-}$ with $\Theta$-classes $a,b,c$ have the same $\tau$-graph $K_3$.
  • Figure 2: Two non-isomorphic daisy cubes with the same $\tau$-graph $K_5$.
  • Figure 3: Daisy cube $A$ with $\Theta$-classes $\{ \mathcal{E}_1, \mathcal{E}_2, \mathcal{E}_3 \}$ such that $A^{\tau} \setminus \{ \mathcal{E}_3 \}$ is not isomorphic to $\bar{A}^{\tau}$.
  • Figure 4: Illustration of Claim 3.

Theorems & Definitions (8)

  • Lemma 3.1
  • Theorem 3.2
  • Corollary 3.3
  • Lemma 4.1
  • Lemma 4.2
  • Theorem 4.3
  • Lemma 4.4
  • Proposition 4.5