Upper bound on the number of edges of an almost planar bipartite graph

TL;DR

This paper establishes tight upper bounds on the number of edges in bipartite graphs that can be drawn in the plane with each edge intersecting at most one other edge, contributing to the understanding of almost planar bipartite graphs.

Contribution

It provides the first tight bounds on edges for almost planar bipartite graphs and constructs examples achieving these bounds.

Findings

Maximum edges are 3v-8 for even v≠6

Maximum edges are 3v-9 for odd v and v=6

Examples matching bounds are constructed for all v≥4

Abstract

Let $G$ be a bipartite graph without loops and multiple edges on $v\ge 4$ vertices, which can be drawn on the plane such that any edge intersects at most one other edge. We prove that such graph has at most $3v-8$ edges for even $v\ne 6$ and at most $3v-9$ edges for odd $v$ and $v=6$. For all $v\ge 4$ examples showing that these bounds are tight are constructed. In the end of paper we discuss a question about drawings of complete bipartite graphs on the plane such that any edge intersects at most one other edge. {\sc Keywords:} topological graphs, planar graphs, bipartite graphs.