Two active NAC-colorings

In our paper, we prove that If a graph $G$ has an injective embedding $\omega:V_G\rightarrow\mathbb{R}^3$ such that for every edge $uv\in E$, the vector $\omega(u)-\omega(v)$ is parallel to one of the four vectors $(1,0,0)$, $(0,1,0)$, $(0,0,1)$, $(-1,-1,-1)$, and all four directions are present, then $G$ is movable. Moreover, there exists an algebraic motion of $G$ with exactly two active NAC-colorings modulo conjugation. Two edges are parallel in the embedding $\omega$ if and only if they receive the same pair of colors in the two active NAC-colorings.

For every graph, an algebraic motion obtained by the lemma is shown here with its two active NAC-colorings. The edges with the same edge lengths are colored by the same colors in the animations.

Graph $Q_1$

Graph $Q_2$

Graph $Q_3$

Graph $Q_4$

Graph $Q_5$

Graph $Q_6$

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