On disjoint crossing families in geometric graphs

A geometric graph is a graph drawn in the plane with vertices represented by points and edges as straight-line segments. A geometric graph contains a (k,l)-crossing family if there is a pair of edge subsets E_1,E_2 such that |E_1| = k and |E_2| = l, the edges in E_1 are pairwise crossing, the edges in E_2 are pairwise crossing, and every edges in E_1 is disjoint to every edge in E_2. We conjecture that for any fixed k,l, every n-vertex geometric graph with no (k,l)-crossing family has at most c_{k,l}n edges, where c_{k,l} is a constant that depends only on k and l. In this note, we show that every n-vertex geometric graph with no (k,k)-crossing family has at most c_kn\log n edges, where c_k is a constant that depends only on k, by proving a more general result which relates extremal function of a geometric graph F with extremal function of two completely disjoint copies of F. We also settle the conjecture for geometric graphs with no (2,1)-crossing family. As a direct application, this implies that for any circle graph F on 3 vertices, every n-vertex geometric graph that does not contain a matching whose intersection graph is F has at most O(n) edges.

💡 Research Summary

The paper investigates extremal edge‑density problems for geometric graphs under a novel forbidden configuration called a ((k,\ell))-crossing family. A geometric graph is a straight‑line drawing of a graph in the plane. A ((k,\ell))-crossing family consists of two edge sets (E_{1}) and (E_{2}) with (|E_{1}|=k) and (|E_{2}|=\ell) such that every pair of edges inside (E_{1}) cross, every pair inside (E_{2}) cross, and no edge of (E_{1}) meets any edge of (E_{2}). The authors conjecture that for any fixed (k,\ell) there is a constant (c_{k,\ell}) so that any (n)-vertex geometric graph avoiding a ((k,\ell))-crossing family has at most (c_{k,\ell}n) edges.

The main contribution is a general reduction theorem that relates the extremal function (f_{F}(n)) of a forbidden geometric subgraph (F) to the extremal function of two vertex‑disjoint copies of (F). Formally, if (F) is a fixed geometric graph and (F\sqcup F) denotes the disjoint union of two copies, then \