1 open statement

• grid_planar

  Every grid graph is planar.

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1 open statement

• planar_iff_kuratowskiFree

  A finite simple graph admits a crossing-free straight-line drawing exactly when it contains no subdivision of K₅ or K₃,₃. This is Kuratowski's theorem together with Fáry's straight-line drawing theorem.

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1 open statement

• ladder_outerplanar

  Every ladder graph is outerplanar.

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1 open statement

• ladder_planar

  Every ladder graph is planar.

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1 open statement

• ladder_seriesParallel

  Every ladder graph is series-parallel.

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1 open statement

• path_outerplanar

  Every path graph is outerplanar.

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1 open statement

• path_planar

  Every path graph is planar.

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1 open statement

• path_tree

  Every path graph is a tree.

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1 open statement

• planar_iff_excludedMinors

  For every finite simple graph, admitting a crossing-free drawing is equivalent to containing neither K₅ nor K₃,₃ as a minor.

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1 open statement

• seriesParallel_planar

  Every series-parallel graph is planar.

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1 open statement

• star_outerplanar

  Every star graph is outerplanar.

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1 open statement

• star_planar

  Every star graph is planar.

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1 open statement

• star_tree

  Every star graph is a tree.

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1 open statement

• tree_outerplanar

  Every tree is outerplanar.

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1 open statement

• tree_planar

  Every tree is planar.

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1 open statement

• triangle_maximalOuterplanar

  Every triangle is maximal outerplanar.

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1 open statement

• triangle_outerplanar

  Every triangle is outerplanar.

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1 open statement

• triangle_planar

  Every triangle is planar.

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1 open statement

• kuratowskiFree_iff_excludedMinors

  A graph contains a subdivision of K₅ or K₃,₃ exactly when it contains K₅ or K₃,₃ as a minor. This special equivalence does not hold for arbitrary forbidden graphs.

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1 open statement

• wall_planar

  Every wall graph is planar.

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1 open statement

• wheel_halin

  Every wheel graph is a Halin graph.

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1 open statement

• wheel_planar

  Every wheel graph is planar.

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1 open statement

• exists_nearLinearTime_randomized_welzlOrder_program

  Near-linear computation of graph Welzl orders (Dreier–Kuske, Theorem 1.3): one randomized word-RAM program, given a member of a graph class whose neighborhood complexity is uniformly at most $c · k$, returns with probability at least $2/3$ an order with crossing number at most $12 · c^2 · log₂(n)^2$, within a constant multiple of $(n+m) log n$ steps.

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1 open statement

• exists_prime_between

  For every $n ≥ 1$ there is a prime $p$ with $n < p ≤ 2n$.

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1 open statement

• exists_odd_prime_between

  For every $n ≥ 2$ there is an odd prime $p$ with $n < p ≤ 2n$.

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