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The aim is to determine the uniqueness of invariant projections for the cycle space of polyhedral graphs.

August 11, 2026Open Access

Invariant Projections on Cycle Spaces of Polyhedral Graphs: An Orbit-Count Formula and a Characterisation of the Regular Solids

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The aim is to determine the uniqueness of invariant projections for the cycle space of polyhedral graphs.
Utilized the orbit-count formula to analyze c(Γ) for convex polyhedra.
Proved duality invariance and established conditions for c(Γ) = 0 corresponding to Platonic solids.
Provided counterexamples and closed formulas for various map types, including chiral toroidal maps.
Derived the orbit formula showing the relationship between automorphism-group orbits and chirality correction.
Established that c(Γ) = 0 only if the graph corresponds to a Platonic solid, based on geometric constraints.
Demonstrated that spherical characterizations do not apply uniformly to toroidal maps or projective planes.

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This manuscript reports on invariant projections of cycle spaces in polyhedral graphs, implying unique characterizations for regular solids.

Dr. Thomas P. Connelly (2026) studied this question.

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This manuscript reports on invariant projections of cycle spaces in polyhedral graphs, implying unique characterizations for regular solids.