Not all partial cubes are $\Theta$-graceful

TL;DR

The paper demonstrates that not all partial cubes are $ heta$-graceful by providing a counterexample, answering a previously open question in graph theory.

Contribution

It presents a specific counterexample of a partial cube that is not $ heta$-graceful, disproving the conjecture that all partial cubes possess this property.

Findings

Counterexample of a non-$ heta$-graceful partial cube

Disproof of the conjecture that all partial cubes are $ heta$-graceful

Clarification of the limitations of $ heta$-gracefulness in partial cubes

Abstract

It is shown that the graph obtained by merging two vertices of two 4-cycles is not a $\Theta$-graceful partial cube, thus answering in the negative a question by Bre\v{s}ar and Klav\v{z}ar from [1], who asked whether every partial cube is $\Theta$-graceful.