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.

翻译：邻接交叉图是一类可被绘制成使得任意两条与同一条边相交的边共享一个公共端点的图。我们证明，具有 $n$ 个顶点的邻接交叉图的边数至多为 $5n-10$。若要求边被绘制为直线段，则该上界变为 $5n-11$。这两个界都是紧的。前一结果也源自Cheong等人近期独立完成的工作\cite{cheong2023weakly}，他们证明了弱扇形图和强扇形图的最大规模相等。通过将此结果与Kaufmann和Ueckerdt\cite{KU22}关于强扇形图规模的界以及Brandenburg\cite{Br20}的结论（即邻接交叉图的最大规模等于扇交叉图的最大规模，而后者又等于弱扇形图的最大规模）相结合，可得到相同的邻接交叉图规模上界。然而，本文提出的证明方法不同、更简洁且更直接。