Sharp volume and face bounds for Hansen polytopes of cographs
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For the Hansen polytope of a nonempty cograph G, with dimension d=|V(G)|+1, are normalized Mahler volume, nonempty-face count and full-flag count bounded below by 1,3ᵈ and 2ᵈd!, respectively? The proposed equality cases are exactly threshold graphs.
Why it matters and what counts as progress
Why it matters: The three equality classifications connect volume, faces and flags of cograph Hansen polytopes to the same threshold-graph extremizers.
What counts as progress: Prove all three sharp bounds and their equality classifications or extend them beyond cographs.
How the problem developed
AI contribution
AI-assisted sharp bounds for Hansen polytopes of cographs
For every nonempty cograph G, its Hansen polytope in dimension d=|V(G)|+1 has normalized Mahler volume at least one, at least 3^d nonempty faces and at least 2^d d! full flags. Each equality holds exactly for threshold graphs.
Sources
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