A sufficient condition for the existence of plane spanning trees on geometric graphs

Let P be a set of n > 2 points in general position in the plane and let G be a geometric graph with vertex set P. If the number of empty triangles uvw in P for which the subgraph of G induced by {u,v,w} is not connected is at most n-3, then G contains a non-self intersecting spanning tree.

💡 Research Summary

The paper investigates the existence of non‑crossing (plane) spanning trees in geometric graphs whose vertices are a set P of n ≥ 3 points in general position in the Euclidean plane. An “empty triangle” uvw is a triangle whose interior contains no other points of P. An empty triangle is called “disconnected” in a geometric graph G if the subgraph induced by {u, v, w} is not connected. Let s(G) denote the number of disconnected empty triangles of G.

The main result (Theorem 1) states that if s(G) ≤ n − 3, then G contains a plane spanning tree. The bound is tight: the complement of the Hamiltonian path on a regular n‑gon, denoted T_c^n, has s(T_c^n) = n − 2 and contains no plane spanning tree.

The proof proceeds by induction on n. For n = 3, 4 the claim is verified directly. For n ≥ 5, the authors select an oriented line L₁ passing through a vertex v₁ and partition P into left and right sets L₁⁻ and L₁⁺ with sizes ⌊(n+1)/2⌋ and ⌈(n+1)/2⌉ respectively. They then employ the classic rotating‑line technique of Erdős, Lovász, Simmons, and Straus to generate all k‑sets of P: the line is rotated around successive points v₂, v₃, …, producing a sequence of oriented lines L_i and corresponding point sets L_i⁻, L_i⁺.

A crucial auxiliary result (Lemma 2) asserts that if three points x, y, z lie in L_i⁺ ∩ L_j⁻ for i < j, then there exist indices k, l with i ≤ k < l < j such that v_k is one of {x, y, z} and the line L_l crosses the triangle xyz. This lemma guarantees that during the rotation process a line eventually intersects any disconnected empty triangle.

The induction step is divided into four exhaustive cases based on the values of s(G_i⁻) and s(G_i⁺) relative to the sizes of the corresponding point sets.

1. Both sides have ≤ |L_i⁻| − 3 and ≤ |L_i⁺| − 3 disconnected triangles. By the inductive hypothesis each side contains a plane spanning tree; their union shares exactly one vertex (the line’s pivot) and yields a plane spanning tree of G.

2. Both sides have exactly |L_i⁻| − 2 and |L_i⁺| − 2 disconnected triangles. In this delicate situation the line L₁ does not intersect any disconnected triangle. Rotating further, let L_j be the first line that does intersect a disconnected triangle Δ = xyz. The vertex v_j must be one of {x, y, z}. Depending on whether the next point v_{j+1} lies on the left or right of L_j, the authors identify an index i > j such that after swapping a single point from one side to the other, the number of disconnected triangles on that side drops by one. This reduction allows the inductive hypothesis to be applied to the modified subgraphs, and the resulting trees are combined to form a spanning tree of G.

3. One side has ≤ |L_i⁻| − 3 while the other has ≥ |L_i⁺| − 2. By considering the line L_m parallel to L₁ with opposite orientation, the authors derive a contradiction unless there exists an intermediate line L_k where the situation switches to case 1 or 2.

4. The symmetric situation of case 3 (left side dense, right side sparse) is handled analogously, with extra care for the even‑n case where the two parallel lines are distinct.

In each scenario the authors either directly apply the induction hypothesis or modify the partition so that the hypothesis becomes applicable. Consequently, a plane spanning tree exists for any geometric graph satisfying s(G) ≤ n − 3.

The paper concludes with a “final remark” presenting another family of examples R_n and R_c^n (where R_c^n has exactly n − 3 disconnected empty triangles) that both admit plane spanning trees, illustrating that Theorem 1 is not a trivial corollary of the earlier Ramsey‑type result by Károlyi et al.

Overall, the work contributes a clean combinatorial condition based on empty‑triangle connectivity, provides a constructive proof via rotating‑line techniques, and establishes the optimality of the bound. It opens avenues for algorithmic implementations that could detect a suitable line and build the spanning tree efficiently, as well as for extending the condition to higher‑dimensional point sets or to other non‑crossing structures.