The maximum size of adjacency-crossing graphs

TL;DR

This paper establishes tight upper bounds on the number of edges in adjacency-crossing graphs, showing they are at most 5n-10 with straight-line edges at 5n-11, and provides a simpler proof for these bounds.

Contribution

It introduces a new, simpler proof for the maximum size bounds of adjacency-crossing graphs, improving understanding of their structural limitations.

Findings

Maximum edges in adjacency-crossing graphs are at most 5n-10.

Straight-line adjacency-crossing graphs have at most 5n-11 edges.

Bounds are tight and match previous results through different proofs.

Abstract

An adjacency-crossing graph is a graph that can be drawn such that every two edges that cross the same edge share a common endpoint. We show that the number of edges in an $n$-vertex adjacency-crossing graph is at most $5n-10$. If we require the edges to be drawn as straight-line segments, then this upper bound becomes $5n-11$. Both of these bounds are tight. The former result also follows from a very recent and independent work of Cheong et al.\cite{cheong2023weakly} who showed that the maximum size of weakly and strongly fan-planar graphs coincide. By combining this result with the bound of Kaufmann and Ueckerdt\cite{KU22} on the size of strongly fan-planar graphs and results of Brandenburg\cite{Br20} by which the maximum size of adjacency-crossing graphs equals the maximum size of fan-crossing graphs which in turn equals the maximum size of weakly fan-planar graphs, one obtains the same bound on the size of adjacency-crossing graphs. However, the proof presented here is different, simpler and direct.