Isomorphic daisy cubes based on their $\tau$-graphs

TL;DR

This paper establishes a characterization of isomorphic daisy cubes via their $ au$-graphs, linking their structure to properties of plane bipartite graphs and providing new proofs for known properties of Fibonacci and Lucas cubes.

Contribution

It introduces a new criterion for daisy cube isomorphism based on $ au$-graphs being forests and connects this to resonance graphs of plane bipartite graphs.

Findings

Daisy cubes with forest $ au$-graphs are isomorphic if their $ au$-graphs are isomorphic.

A daisy cube with edges corresponds to the resonance graph of a plane bipartite graph.

Provides alternative proofs for properties of Fibonacci and Lucas cubes.

Abstract

We prove that if $A$ and $B$ are daisy cubes whose $\tau$-graphs are forests, then $A$ and $B$ are isomorphic if and only if their $\tau$-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 $\tau$-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.