The maximum size of adjacency-crossing graphs
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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.<cit.> 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<cit.> on the size of strongly fan-planar graphs and results of Brandenburg<cit.> 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.
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