4 problems

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Connelly's conjecture on optimality of triangulated disc packings
Let a finite set of discs be given. A packing is triangulated saturated if it is both triangulated and saturated, meaning that its contact structure is a triangulation and no furth…
Conjecturesolveddisc-packings geometric-optimizationarXiv posted 2 days ago
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Conjecture on phase separation in the remaining binary disc packing cases
Phase-separation conjecture. The second case should be very similar to the first, with phase separation only when there is an excess of small discs. The last two cases should be le…
Conjectureopencombinatorics density disc-packingsarXiv posted 2 days ago
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The saturation conjecture for densest packings of discs with finitely many radii
Consider packings by $n$ different given discs. A packing is saturated if no further disc can be added. Saturation conjecture. If the densest compact packing is saturated, then the…
Conjectureopendisc-packings geometric-combinatorics packing-densityarXiv posted 2 days ago
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Conjecture on the number of compact disc packings
Enumeration conjecture. The number of compact disc packings of order $k$ is equal to the $(k+1)^{\text{st}}$ term in the OEIS sequence A086759.
Conjectureopencombinatorics disc-packings geometry-of-numbersarXiv posted 2 days ago

Connelly's conjecture on optimality of triangulated disc packings