A new proof of Vázsonyi's conjecture

We present a self-contained proof that the number of diameter pairs among n points in Euclidean 3-space is at most 2n − 2. The proof avoids the ball polytopes used in the original proofs by Grünbaum, Heppes and Straszewicz. As a corollary we obtain that any three-dimensional diameter graph can be embedded in the projective plane. Let S be a set of n points of diameter D in Rd. Define the diameter graph on S by joining all diameters, i.e., point pairs at distance D. The following theorem was conjectured by Vázsonyi, as reported in [2]. It was subsequently independently proved by Grünbaum [3], Heppes [4] and Straszewicz [7]. Theorem 1. The number of edges in a diameter graph on n ≥ 4 points in R3 is at most 2n− 2. All three proofs (see [6, Theorem 13.14]) use the ball polytope obtained by taking the intersection of the balls of radius D centred at the points. However, these ball polytopes do not behave the same as ordinary polytopes. In particular, their graphs need not be 3connected, as shown by Kupitz, Martini and Perles in [5], where a detailed study of the ball polytopes associated to the above theorem is made. The proof presented here avoids the use of ball polytopes. Theorem 2. Any diameter graph in R3 has a bipartite double covering that has a centrally symmetric drawing on the 2-sphere. In fact, each point x ∈ S will correspond to an antipodal pair of points xr and xb on the sphere, with xr coloured red and xb blue. Each edge xy of the diameter graph will correspond to two antipodal edges xryb and xbyr on the sphere, giving a properly 2-coloured graph on 2n vertices. The drawing will be made such that no edges cross. By Euler’s formula there will be at most 4n− 4 edges, hence at most 2n− 2 edges in the diameter graph. By identifying opposite points of the sphere we further obtain: This material is based upon work supported by the South African National Research Foundation. 1 2 KONRAD J. SWANEPOEL Corollary 3. Any diameter graph in R3 can be embedded in the projective plane such that all odd cycles are noncontractible. Therefore, any two odd cycles intersect, and we regain the following theorem of Dol’nikov [1]: Corollary 4. Any two odd cycles in a diameter graph on a finite set in R3 intersect. Proof of Theorem 2. Without loss we assume from now on that D = 1. Let S2 denote the sphere in R3 with centre the origin and radius 1. We may repeatedly remove all vertices of degree at most 1 in the diameter graph. Since such vertices can easily be added later, this is no loss of generality. For each x ∈ S, let R(x) be the intersection of S2 with the cone generated by {y− x : xy is a diameter}. Each R(x) is a convex spherical polygon with great circular arcs as edges. (If x has degree 2 then R(x) is an arc). Colour R(x) red and B(x) := −R(x) blue. Assume for the moment the following two properties of these polygons: Lemma 1. If x 6= y, then R(x) and R(y) are disjoint. Lemma 2. If R(x) and B(y) intersect, then xy is a diameter and R(x) ∩ B(y) = {y− x}. For each x ∈ Swe choose any xr in the interior of R(x) and let xb = −xr. (If R(x) is an arc we let xr be in its relative interior.) Draw arcs inside R(x) from xr to all the vertices of R(x), as well as antipodal arcs from xb to the vertices of B(x). This gives a centrally symmetric drawing of a 2-coloured double covering of the diameter graph. By Lemmas 1 and 2 no edges cross, and the theorem follows. The following proofs of Lemmas 1 and 2 are dimension independent, which gives a double covering on Sd−1 of any diameter graph in Rd. Lemma 3. Let x1, . . . , xk and ∑ k i=1 λixi be unit vectors in R d, with all λi ≥ 0. Suppose that for some y ∈ R d, ‖y− xi‖ ≤ 1 for all i = 1, . . . , k. Then ‖y− ∑ i=1 λixi‖ ≤ 1. Proof. By the triangle inequality, 1 ≤ ‖ k ∑ i=1 λixi‖ ≤ k ∑ i=1 λi. (1) Expanding ‖y− xi‖ 2 ≤ 1 by inner products, − 2 〈xi, y〉 ≤ −‖y‖ . (2) A NEW PROOF OF VÁZSONYI’S CONJECTURE 3