The article proves and are toroidal penny graphs.
problem Optimal sphere packing on torus.
method Analyzing connections between planar graphs, penny graphs, and toroidal penny graphs.
result and are toroidal penny graphs.
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Researchers found all embeddings of Kuratowski graphs on a double torus.
problem Characterizing embeddings of Kuratowski graphs and on the double torus.
method Constructive approach using Burnside's Lemma and automorphism groups.
result 14 orientable and 17 non-orientable 2-cell embeddings of on the double torus.