Graphs with Large Girth Not Embeddable in the Sphere

In 1972, M. Rosenfeld asked if every triangle-free graph could be embedded in the unit sphere S in such a way that two vertices joined by an edge have distance more than √ 3 (i.e. distance more than 2π/3 on the sphere). In 1978, D. Larman [4] disproved this conjecture, constructing a triangle-free graph for which the minimum length of an edge could not exceed p 8/3. In addition, he conjectured that the right answer would be √ 2, which is no better than the class of all graphs. Larman’s conjecture was independently proved by M. Rosenfeld [7] and V. Rődl [6]. In this last paper it was shown that no bound better than √ 2 can be found for graphs with arbitrarily large odd girth. We prove in this paper that this is still true for arbitrarily large girth. We discuss then the case of triangle-free graphs with linear minimum degree. Fix a real α > 0 and an integer d ≥ 1. The Borsuk graph Bor(d, (1 + α)π/2) is the (infinite) graph defined on the d-dimensional unit sphere where two points are joined by an edge if and only if the distance on the sphere is more than (1 + α)π/2. A graph is α-spherical if it is a subgraph of Bor(d, (1 + α)π/2) for some d. D. Larman in [4] asked if for every α > 0, there exists a triangle-free graph which is not α-spherical. The problem was popularized by P. Erdős, and was independently proved by M. Rosenfeld [7] and V. Rődl [6]. See [5] for a survey. In [6] was also proved that for every α > 0, there exists a graph with arbitrarily large odd-girth which is not α-spherical. We generalize this result to graphs with arbitrarily large girth. Revisiting the problem twenty years later is much easier: one reason is that the probabilistic method is now widely spread, and the other reason is that the work of Goemans and Williamson on max-cut [3] highlighted the close relationship between sphere-embedding of graphs and cuts. Here is the key-observation: Lemma 1 If G is α-spherical there exists a cut of G which has at least (1 + α)m/2 edges, where m is the total number of edges. Proof. Embed G in some S in such a way that every edge has spherical length at least (1 + α)π/2. Observe that a random hyperplane cut an edge of G with probability (1 + α)/2. By double counting, there is some hyperplane which cuts at least (1 + α)m/2 edges. This is our cut. Now the proof is almost finished, since a graph G satisfying Lemma 1 is certainly far from being random. And indeed, Erdős’ random graphs with large girth and large χ, when they have enough vertices, are not α-spherical. This is the next result: Lemma 2 For every α > 0 and every integer k, there exists a graph G, with girth at least k, in which every cut has less than (1 + α)m/2 edges, where m is the number of edges of G. Proof. We consider for this random graphs on n vertices with independently chosen edges with probability p. We first want to bound the size of a maximum cut. Let A be a subset of vertices. Let X denotes the number of edges between A and its complement. The