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On the existence of paradoxical motions of generically rigid graphs on the sphere

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Pith's one-line read A graph has a flexible assignment of spherical edge lengths if and only if it admits a NAP-coloring, and K3,3 has exactly three proper spherical motions: two Dixon-type motions and one new constant diagonal angle motion.

arxiv 1908.00467 v2 pith:HKO66VYJ submitted 2019-08-01 math.CO cs.ROmath.AGmath.MG

Matteo Gallet , Georg Grasegger , Jan Legerský , Josef Schicho This is my paper · ORCID
classification math.COcs.ROmath.AGmath.MG
keywords spheregraphsedgesexistencegraphlengthsmotionsrealizations
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The reading

A framework is a graph whose edges are bars of fixed length; a realization places the vertices on a surface so those bars fit. On the plane, it was already known that a graph can be moved for some choice of bar lengths exactly when its edges admit a special two-coloring called a NAC-coloring. This paper proves the analogous statement on the sphere, but with a different and stricter coloring condition, called NAP: no four-vertex path may have colors that alternate along its three edges.

The proof has two halves. One half is constructive: if a NAP-coloring exists, the vertices that touch both colors are placed at the two poles of the sphere, and the blue subgraph is rotated around the polar axis while the red subgraph stays still. The other half uses algebraic geometry: realizations on the sphere are translated into points of a moduli space of curves, and a flexible framework must degenerate at the boundary of this moduli space, producing a colored cut called a bond that is exactly a NAP-coloring.

The second part of the paper is a case study of the smallest generically rigid graph that admits paradoxical motion on the sphere, the complete bipartite graph K3,3. Using a careful classification of how its 4-vertex subgraphs move, the authors show there are only three possible motions with no coincident or antipodal vertices: two spherical versions of Dixon's classical planar motions, and a third, new motion in which the angle between two diagonals of a 4-cycle stays constant. This new motion is described by an explicit parametrization.

Extended reading notes

Core claim

Theorem 3.14: "Let G be a connected graph without multiedges, or self-loops. Then G admits a flexible assignment of edge lengths on the sphere if and only if it admits a NAP-coloring." If the paper is correct, the combinatorial NAP-coloring condition completely determines which graphs can be flexed on the sphere for some choice of edge lengths, including generically rigid Laman graphs.

Load-bearing premise

The only-if direction of Theorem 3.14 and all of Section 4 depend on the moduli-space reduction imported from the same-author paper [GGS19]: realizations of an n-vertex graph on the sphere up to SO(3) correspond to points in the compactified moduli space M̄_{0,2n}, and the spherical distance is read off as a cross-ratio of four lift points on a conic (Section 2, Definition 2.5 and surrounding text). That reduction is cited rather than reproved or machine-checked, so the bond and NAP-coloring conclusions inherit any error in it. Separately, the K3,3 classification assumes the completeness of the spherical quadrilateral classification of [GS88a, GS88b] and the correctness of the 26-case degree-table search (Lemma 4.18, Proposition 4.21).

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Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central theorem rests on the GGS19 moduli-space reduction (same-author prior work) and on the bond-to-cut arguments; the K3,3 classification additionally rests on a finite case search and on the prior spherical quadrilateral classification. No numeric constants are fitted to data; the only parameters (a and e in the new motion) satisfy a derived algebraic relation and are not fitted.

assumptions (5)
  • domain assumption The GGS19 reduction: realizations on the sphere up to SO(3) correspond to points in M̄_{0,2n}; spherical distance is the cross-ratio of lifts; the map η: M̄_{0,2n} -> CM_n is surjective.
    Imported from the same-author paper [GGS19]; foundational for the bond argument and for computing fiber dimensions, but not proven or machine-checked in this manuscript (Section 2 and Lemma 3.6).
  • standard math Keel's intersection theory of M̄_{0,n}, including quadratic relations in the Chow ring, is used to compute divisor intersection numbers in Section 4.1 (for example, Z·Dou = 0).
    Standard background from [Kee92], used to derive Table 1 for general, deltoid, rhomboid, and lozenge cases.
  • domain assumption The classification of movable spherical quadrilaterals (general, odd/even deltoid, rhomboid, lozenge) from [GS88a, GS88b] is complete for real proper motions.
    Definition 4.7 asserts these five families are exhaustive, relying on prior work and on the hypothesis of real proper motions; this completeness is not proved in the paper.
  • ad hoc to paper A finite case analysis shows that only four type tables are allowed after inspecting the 26 degree-table orbits.
    Lemma 4.18 and the discussion after Proposition 4.21 describe a computer-assisted enumeration, but no code, system name, or output is provided; it is load-bearing for the K3,3 classification.
  • domain assumption Generic rigidity on the sphere coincides with Laman graphs, as proved in [EJN+19].
    Used to frame the 'paradoxical motion' result for generically rigid graphs; not needed for the proof of Theorem 3.14 itself, but it motivates the problem.

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Pith. "Pith review of On the existence of paradoxical motions of generically rigid graphs on the sphere." pith.science (2026). https://pith.science/paper/HKO66VYJ

@misc{pith2026190800467,
  author       = {Pith},
  title        = {Pith review of: On the existence of paradoxical motions of generically rigid graphs on the sphere},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HKO66VYJ}},
  note         = {Machine review of arXiv:1908.00467}
}
abstract

We interpret realizations of a graph on the sphere up to rotations as elements of a moduli space of curves of genus zero. We focus on those graphs that admit an assignment of edge lengths on the sphere resulting in a flexible object. Our interpretation of realizations allows us to provide a combinatorial characterization of these graphs in terms of the existence of particular colorings of the edges. Moreover, we determine necessary relations for flexibility between the spherical lengths of the edges. We conclude by classifying all possible motions on the sphere of the complete bipartite graph with $3+3$ vertices where no two vertices coincide or are antipodal.

Figures

Figures reproduced from arXiv: 1908.00467 by the authors.

Figure 1
Figure 1. Realizations of graphs on the sphere: the unique graph with 4 vertices and 5 edges (left and middle) and the bipartite complete graph K3,3 on the right. Although the combinatorial characterization we obtain applies to all graphs, it is of particular interest in the case of generically rigid graphs. These are graphs such that for a general realization on the sphere there are only finitely many essentially distinct re… view at source ↗
Figure 2
Figure 2. On the left, an alternating path. On the right, a path which is not alternating [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. An example of a NAP-coloring of the complete bipar￾tite graph K3,2. Remark 3.12. Every NAP-coloring is a NAC-coloring. Proposition 3.13. Let (I, J) be a cut induced by a bond of a graph G. The coloring εI,J from Definition 3.8 is a NAP-coloring if and only if there exists a non-edge {c, d} for which Pc, Qc ∈ I and Pd, Qd ∈ J. Proof. Assume that there exists such a non-edge {c, d}. Let us suppose that there is an alt… view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: The NAP-colorings on the right induce flexible assign￾ments: the two (respectively, three) vertices incident to both blue and red edges are mapped to two antipodal points on the sphere. For the second graph, this leads to identification of two points: the vertex on top…
Figure 5
Figure 5. Figure 5: Smallest minimally rigid graph with a NAP-coloring. In the corresponding flexible assignment, the leftmost and the rightmost vertex are sent to the same point or to two antipodal points on the sphere. We conclude this section by pointing out how our setup allows us to …
Figure 6
Figure 6. Figure 6: The two motions of the complete graph K3,3 discovered by Dixon. In the motion on the left, three points are moving on the x-axis, and three points are moving on the y-axis. The right is a symmetric motion of K4,4 consisting of the vertices of two rectangles sharing the…
Figure 7
Figure 7. Figure 7: The complete bipartite graph K3,3 has exactly 6 NAP-colorings. We start by defining what we mean by “motion”. Here we make clear that we con￾sider real assignments of edge lengths, and real points on the sphere. Furthermore, we focus on motions for which no two vertice…
Figure 8
Figure 8. Figure 8: The three involutions describing a Dixon 2 motion [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: An odd deltoid (left) and two degenerate deltoids where vertices 1 and 3 coincide (middle) or are antipodal (right). Vertex 2 may move along the dashed circle. deg r 56 i = 2 for i even. Moreover, for all realizations (R1, . . . , R4) ∈ [PITH_FULL_IMAGE:figures/full_f…
Figure 10
Figure 10. Figure 10: Three rhomboids: The rhomboid on the left has a symmetry plane, which intersects the sphere in the green circle. The rhomboid in the middle has a symmetry line, which meets the sphere in the intersection of the two diagonals. The rhomboid on the right is obtained from…
Figure 11
Figure 11. Figure 11: Two lozenges with four equal side lengths. The diag￾onals are orthogonal and bisect each other. Remark 4.8. A lozenge is not a special case of a rhomboid: in fact, in a rhomboid the configuration curve has only two components, and this excludes the situation where all…
Figure 12
Figure 12. Figure 12: Visualization of four realizations of K3,3 during a con￾stant diagonal angle motion. The diagonals (in green) and their poles on the silhoutte circle remain fixed. The cosine of the orange edge between the poles is 3 4 . The angles of the other orange edges are π 2 . …

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