The wheel-graph adjacency-polytope volume conjecture

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Let WN=K1∨CNW_N=K_1\vee C_N be the wheel graph obtained by joining a single vertex to the cycle CNC_N, and let NVol⁡(∇WNPQ⁡)\operatorname{NVol}(\nabla^{\operatorname{PQ}}_{W_N}) denote the normalized volume of its type-PQ adjacency polytope. Wheel-graph volume conjecture. For all N≥3N\geq 3,

NVol⁡(∇WNPQ⁡)=3N−2N+1.\operatorname{NVol}(\nabla^{\operatorname{PQ}}_{W_N})=3^N-2^N+1.

This gives a conjectural closed formula for the normalized volumes of wheel graphs, a planar class that is not generally outerplanar. The source provides no proof or resolution, so the conjecture remains open.

References

Primary source

Robert Davis and Tianran Chen, “Computing Volumes of Adjacency Polytopes via Draconian Sequences”, arXiv:2007.11051 (2022).

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