A note on the sizes of bipartite 1-planar graphs

TL;DR

This paper establishes a tight upper bound on the number of edges in bipartite 1-planar graphs with a specific 1-disk OX drawing configuration, advancing understanding of their size limitations.

Contribution

It proves a maximum edge bound for bipartite 1-planar graphs with 1-disk OX drawings, resolving an open problem and confirming the bound's tightness.

Findings

Maximum edges in bipartite 1-planar graphs with 1-disk OX drawings is 2|V(G)|+|X|-6.

The proven upper bound is tight, meaning it can be achieved.

The result advances the understanding of size constraints in bipartite 1-planar graphs.

Abstract

A graph is 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. Let G be a bipartite 1-planar graph with partite sets X and Y. A 1-disk OX drawing of G is a 1-planar drawing such that all vertices of X lie on the boundary of O and all vertices of Y and all edges of G locate in the interior of O, where O is a disk on the plane. The concept was first proposed by Huang, Ouyang and Dong when they solved a conjecture about the edge density of bipartite 1-planar graphs. Additionally, they presented a problem of determining the maximum number of edges in a bipartite graph with a 1-disk OX drawing. In this paper, we solve this problem and prove that every bipartite graph G which has a 1-disk OX drawing has at most 2|V(G)|+|X|-6 edges. Moreover, we demonstrate that this upper bound is tight.