Crossing lemma for the odd-crossing number

TL;DR

This paper generalizes the crossing lemma to odd-crossing numbers, establishing upper bounds for edges in 1-odd-planar graphs and related classes, thus advancing understanding of graph crossing properties.

Contribution

It introduces the concept of 1-odd-planar graphs, proves their maximum edge count, and improves the crossing lemma constant for graphs with odd-crossing constraints.

Findings

1-odd-planar graphs have at most 5n-9 edges.

Improved the crossing lemma constant for odd-crossing numbers.

Provided bounds for k-odd-planar graphs.

Abstract

A graph is $1$-planar, if it can be drawn in the plane such that there is at most one crossing on every edge. It is known, that $1$-planar graphs have at most $4n-8$ edges. We prove the following odd-even generalization. If a graph can be drawn in the plane such that every edge is crossed by at most one other edge {\em an odd number of times}, then it is called 1-odd-planar and it has at most $5n-9$ edges. As a consequence, we improve the constant in the Crossing Lemma for the odd-crossing number, if adjacent edges cross an even number of times. We also give upper bound for the number of edges of $k$-odd-planar graphs.