Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Almási, Nóra, Tóth, Géza
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223031110