Crossing Number Calculator

min edge crossings cr(G)

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About Crossing Number Calculator

A crossing number calculator computing cr(G): minimum edge crossings in any planar drawing. cr(G)=0 iff planar. cr(K_n)≈n⁴/64 (Guy's conjecture). cr(K_{m,n})≈(m²-m)(n²-n)/16 (Zarankiewicz). NP-hard in general. Client-side.

Crossing Number Calculator Features

  • cr(G) value
  • Planar check
  • Guy conjecture
  • Zarankiewicz
  • Common graphs
Crossing number cr(G): minimum crossings in any drawing. cr=0 iff planar. Guy's conjecture: cr(K_n) = Z(n) = ⌊n/2⌋⌊(n-1)/2⌋⌊(n-2)/2⌋⌊(n-3)/2⌋/4. Verified for n≤12. NP-hard to compute.

How to Use

Select graph:

  • cr: Crossing number
  • Planar: cr=0?
  • Conjecture: Bounds

Guy's Conjecture

cr(K_n) = Z(n) = ⌊n/2⌋⌊(n-1)/2⌋⌊(n-2)/2⌋⌊(n-3)/2⌋/4. Verified for n≤12. Z(5)=1, Z(6)=3, Z(7)=9, Z(8)=18, Z(9)=36. Related to the VLSI chip layout problem.

Applications

VLSI chip design: minimize wire crossings. Graph visualization: cleaner diagrams. Network layout: reduce interference. The crossing lemma: cr(G) ≥ |E|³/(33.75|V|²) for |E|≥4|V|.

Step-by-Step Instructions

  1. 1Select graph.
  2. 2Compute cr(G).
  3. 3Check planarity.
  4. 4Apply bounds.
  5. 5Compare drawings.

Crossing Number Calculator — Frequently Asked Questions

Is crossing number NP-hard?+

Yes! Even determining cr(G)=0 is in P (planarity testing), but cr(G)≤k is NP-complete for variable k. Fixed k: FPT algorithm exists. In practice: heuristics and bounds are used.

What's the crossing lemma?+

If |E|≥4|V|, then cr(G) ≥ |E|³/(33.75|V|²). This probabilistic result (Ajtai-Chvátal-Newborn-Szemerédi) is crucial in combinatorial geometry. It proves the Szemerédi-Trotter theorem on point-line incidences.

What's cr(K_5)?+

cr(K_5) = 1. K_5 is minimally non-planar: removing any edge makes it planar. One crossing suffices. cr(K_6)=3, cr(K_7)=9. The crossing number grows roughly as n⁴/64.

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