Reference. Fulek et al.: Adjacent crossings do matter [cn-0049]
Paper reference 7. Radoslav Fulek, Michael J. Pelsmajer, Marcus Schaefer, and Daniel Štefankovič. Adjacent crossings do matter. Journal of Graph Algorithms and Applications 16(3) (2012), 759–782. Source. Archived PDF.
Section 1, p. 760, distinguishes prohibiting adjacent crossings from allowing them without counting them. The first restriction preserves the ordinary crossing number [cn-0005], as in normalization [cn-0009]. The second gives the independent crossing number [cn-0066], and the authors explicitly leave equality with the ordinary crossing number open, even for complete graphs. Their separations concern odd and algebraic crossing-number variants; they do not provide a counterexample to \(\operatorname {cr}=\operatorname {cr}_-\).
For a later statement of the complete-graph question, see Schaefer's July 2026 survey [cn-0067], Question 2, p. 103. The scope of the September 2026 bipartite argument is explained at its lower-bound step [cn-0035].
- Proof dependencies
- None.
- Provenance
- Bibliographic entry from the pinned paper; abstract and Section 1, pp. 759–760, consulted on 9 October 2026.