On the Number of Edges of Fan-Crossing Free Graphs

TL;DR

This paper establishes tight bounds on the maximum number of edges in fan-crossing free graphs, specifically for 2-fan-crossing free graphs, and explores generalizations to monotone graph properties.

Contribution

It provides the first tight bounds for 2-fan-crossing free graphs and extends the analysis to graphs with higher fan-crossing restrictions and monotone properties.

Findings

Maximum edges in 2-fan-crossing free graphs is 4n-8.

Straight-edge 2-fan-crossing free graphs have at most 4n-9 edges.

For k > 2, the maximum number of edges is bounded by 3(k-1)(n-2).

Abstract

A graph drawn in the plane with n vertices is k-fan-crossing free for k > 1 if there are no k+1 edges $g,e_1,...e_k$, such that $e_1,e_2,...e_k$ have a common endpoint and $g$ crosses all $e_i$. We prove a tight bound of 4n-8 on the maximum number of edges of a 2-fan-crossing free graph, and a tight 4n-9 bound for a straight-edge drawing. For k > 2, we prove an upper bound of 3(k-1)(n-2) edges. We also discuss generalizations to monotone graph properties.