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The study aims to improve existing lower bounds for the crossing number of dense k-planar graphs.

Improving the Crossing Lemma by Characterizing Dense 2-Planar and 3-Planar Graphs

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The study aims to improve existing lower bounds for the crossing number of dense k-planar graphs.
Analyzed 2-planar and 3-planar graphs with constraints on local configurations.
Derived new bounds for crossing numbers based on edge and vertex density.
Applied a probabilistic proof for enhanced constants in the crossing lemma.
Achieved the bound cr(G) ≥ 37/9m - 155/9(n-2) for 5n < m ≤ 6n.
Achieved stronger bounds cr(G) ≥ 5m - 203/9(n-2) for m > 6n.
Improved the constant in the Crossing Lemma to c > 1/27.48 from 1/29.

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Randomized trial improves crossing number bounds in dense k-planar graphs, indicating new analytical tools.

Büngener et al. (2026) studied this question.

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