On the sizes of bipartite 1-planar graphs

TL;DR

This paper proves a conjecture about the maximum number of edges in bipartite 1-planar graphs, establishing tighter bounds based on the sizes of the partite sets, thus advancing understanding of their structural limitations.

Contribution

It confirms a conjecture on edge bounds for bipartite 1-planar graphs, providing a more general and weaker condition than previously considered.

Findings

Established that $m \,\le\, 2n + 4x - 12$ for bipartite 1-planar graphs with partite sets of sizes $x$ and $y$

Extended the known bounds to weaker conditions on the sizes of partite sets

Resolved an open conjecture in the study of bipartite 1-planar graphs.

Abstract

A graph is called $1$-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 $n$ ($\ge 4$) vertices and $m$ edges. Karpov showed that $m\le 3n-8$ holds for even $n\ge 8$ and $m\le 3n-9$ holds for odd $n\ge 7$. Czap, Przybylo and \u{S}krabul\'{a}kov\'{a} proved that if the partite sets of $G$ are of sizes $x$ and $y$, then $m\le 2n+6x-12$ holds for $2\leq x\leq y$, and conjectured that $m\le 2n+4x-12$ holds for $x\ge 3$ and $y\ge 6x-12$. In this paper, we settle their conjecture and our result is even under a weaker condition $2\le x\le y$.