In 1899, Dixon presented two constructions of proper flexible labelings of $K_{4,4}$. More than a century later, Walter and Husty proved that these two motions are the only possible ones.
The first Dixon’s construction works for any bipartite graph: the vertices from one partion set are placed on the $x$-axis and the rest on the $y$-axis.
Dixon I - $K_{3,3}$
There is also a collision free linkage that models this motion:
Dixon II - $K_{4,4}$
The second construction is suitable only for $K_{4,4}$ and its subgraphs. Assume the we have two rectagles with the same intersection of diagonals and such that their edges are either parallel or orthogonal to each other. Now, each partition set is mapped to the vertices of the rectangles.
