TOPICS
Search

Norine's Antipodal-Coloring Conjecture

Norine's antipodal-coloring conjecture (Norine 2008) states that, for every integer n>=2, every red-blue coloring of the graph edges of the hypercube graph Q_n in which antipodal graph edges have opposite colors contains a monochromatic graph path joining some graph vertex to its antipode.

Writing a graph vertex as x=(x_1,...,x_n) in {0,1}^n, the coordinate-complement involution is

A_n(x_1,...,x_n)=(1-x_1,...,1-x_n).

Thus the antipode of the graph edge xy is the graph edge A_n(x)A_n(y).

Wu and Yang (2026) proved the conjecture for all n using a chain-level Borsuk-Ulam theorem obstruction. A hypothetical counterexample gives an antipodally equivariant, augmentation-preserving chain map from the cellular chains of the boundary of a cube to polyhedral chains on a sphere, and the obstruction rules out this map. The stronger version requiring the monochromatic graph path to be a graph geodesic remains open.

Wu and Yang (2026) state that GPT-5.6 Sol Ultra assisted in generating the central proof idea and Codex using GPT-5.6 Sol assisted in drafting the manuscript. As of Sep. 30, 2026, independent review had not been reported.

See also

Antipodal Graph, Borsuk-Ulam Theorem, Graph Coloring, Hypercube Graph

Explore with Wolfram|Alpha

References

Norine, S. "Edge-Antipodal Colorings of Cubes." Open Problem Garden, posted by M. DeVos, Oct. 6, 2008. https://www.openproblemgarden.org/op/edge_antipodal_colorings_of_cubes.Wu, H. and Yang, N. "A Chain-Level Borsuk-Ulam Obstruction Proof of Norine's Antipodal-Coloring Conjecture." 21 Jul 2026. https://arxiv.org/abs/2607.19276.

Cite this as:

Weisstein, Eric W. "Norine's Antipodal-Coloring Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NorinesAntipodal-ColoringConjecture.html

Subject classifications