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PreprintCC BY 4.0arXiv

Improvement of the Crossing Lemma for the odd-crossing number

  • Nóra Almási
  • Géza Tóth
Open at sourcePDFDOICite

Abstract

A graph is called $1$-planar if it can be drawn in the plane so that every edge is crossed by at most one other edge. It is known that the maximum number of edges of $1$-planar graphs is 4n−8. We prove the following generalization. A graph is $1$-odd-planar if it can be drawn in the plane so that every edge is crossed {\em an odd number of times} by at most one other edge. We show that the maximum number of edges of $1$-odd-planar graphs is also 4n−8. As a consequence, we improve the constant coefficient in the Crossing Lemma for the odd-crossing number.

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Cite

BibTeXRIS
BibTeX
@misc{almasi2026improvement,
  title = {{Improvement of the Crossing Lemma for the odd-crossing number}},
  author = {Nóra Almási and Géza Tóth},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25465},
  eprint = {2610.25465},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25465},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Improvement of the Crossing Lemma for the odd-crossing number
AU  - Nóra Almási
AU  - Géza Tóth
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2610.25465
UR  - https://arxiv.org/abs/2610.25465
AB  - A graph is called $1$-planar if it can be drawn in the plane so that every edge is crossed by at most one other edge. It is known that the maximum number of edges of $1$-planar graphs is 4n−8. We prove the following generalization. A graph is $1$-odd-planar if it can be drawn in the plane so that every edge is crossed {\em an odd number of times} by at most one other edge. We show that the maximum number of edges of $1$-odd-planar graphs is also 4n−8. As a consequence, we improve the constant coefficient in the Crossing Lemma for the odd-crossing number.
ER  -