Not all partial cubes are $\Theta$-graceful
Nathann Cohen, Matjaž Kovše · arXiv (Cornell University) · 2018
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.