On an extremal problem in the class of 1-planar graphs

TL;DR

This paper investigates the maximum number of edges in bipartite 1-planar graphs with prescribed partite set sizes, revealing that the edge count approaches 2n for highly unbalanced graphs, contrasting with the known 3n-8 bound for balanced ones.

Contribution

It establishes new bounds on the maximum size of bipartite 1-planar graphs based on the imbalance of partite sets, highlighting the asymptotic behavior for unbalanced cases.

Findings

Maximal bipartite 1-planar graphs with nearly balanced parts have close to 3n-8 edges.

Unbalanced bipartite 1-planar graphs have maximum edges approaching 2n.

Edge count depends significantly on the size disparity between partite sets.

Abstract

A graph $G=(V,E)$ is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. In this paper, we study bipartite $1$-planar graphs with prescribed numbers of vertices in partite sets. Bipartite 1-planar graphs are known to have at most $3n-8$ edges, where $n$ denotes the order of a graph. We show that maximal-size bipartite $1$-planar graphs which are almost balanced have not significantly fewer edges than indicated by this upper bound, while the same is not true for unbalanced ones. We prove that maximal possible sizes of bipartite $1$-planar graphs whose one partite set is much smaller than the other one tends towards $2n$ rather than $3n$. In particular, we prove that if the size of the smaller partite set is sublinear in $n$, then $|E|=(2+o(1))n$, while the same is not true otherwise.