There are lots of interesting invariants of a graph which bound its chromatic number! Most famous is the Lovász number, which asks, roughly: I attach vectors v_x to each vertex x such that v_x and v_y are orthogonal whenever x and y are adjacent, I try to stuff all those vectors into a small cone, the half-angle of the cone tells you the Lovász number, which is bigger and bigger as the smallest cone gets closer and closer to a hemisphere.

Here’s an equivalent formulation: If G is a graph and V(G) its vertex set, I try to find a function f: V(G) -> R^d, for some d, such that

|f(x) – f(y)| = 1 whenever x and y are adjacent.

This is called a *unit distance embedding*, for obvious reasons.

The *hypersphere* number t(G) of the graph is the radius of the smallest sphere containing a unit distance embedding of G. Computing t(G) is equivalent to computing the Lovász number, but let’s not worry about that now. I want to generalize it a bit. We say a finite sequence (t_1, t_2, t_3, … ,t_d) is *big enough* for G if there’s a unit-distance embedding of G contained in an ellipsoid with major radii t_1^{1/2}, t_2^{1/2}, .. t_d^{1/2}. (We could also just consider infinite sequences with all but finitely many terms nonzero, that would be a little cleaner.)

Physically I think of it like this: the graph is trying to fold itself into Euclidean space and fit into a small region, with the constraint that the edges are rigid and have to stay length 1.

Sometimes it can fold a lot! Like if it’s bipartite. Then the graph can totally fold itself down to a line segment of length 1, with all the black vertices going to one end and the white vertices going to the other. And the big enough sequences are just those with some entry bigger than 1.

On the other hand, if G is a complete graph on k vertices, a unit-distance embedding has to be a simplex, so certainly anything with k of the t_i of size at least 1-1/k is big enough. (Is that an if and only if? To know this I’d have to know whether an ellipse containing an equilateral triangle can have a radius shorter than that of the circumcircle.)

Let’s face it, it’s confusing to think about ellipsoids circumscribing embedded graphs, so what about instead we define t(p,G) to be the minimum value of the L^p norm of (t_1, t_2, …) over ellipsoids enclosing a unit-distance embedding of G.

Then a graph has a unit-distance embedding in the plane iff t(0,G) <= 2. And t(oo,G) is just the hypersphere number again, right? If G has a k-clique then t(p,G) >= t(p,K_k) for any p, while if G has a k-coloring (i.e. a map to K_k) then t(p,G) <= t(p,K_k) for any n. In particular, a regular k-simplex with unit edges fits into a sphere of squared radius 1-1/k, so t(oo,G) < 1-1/k.

So… what’s the relation between these invariants? Is there a graph with t(0,G) = 2 and t(oo,G) > 4/5? If so, there would be a non-5-colorable unit distance graph in the plane. But I guess the relationship between these various “norms” feels interesting to me irrespective of any relation to plane-coloring. What is the max of t(oo,G) with t(0,G)=2?

The intermediate t(p,G) all give functions which upper-bound clique number and lower-bound chromatic number; are any of them interesting? Are any of them easily calculable, like the Lovász number?

**Remarks:**

- I called this post “What is the Lovász number of the plane?” but the question of “how big can t(oo,G) be if t(0,G)=2”? is more a question about finite subgraphs of the plane and their Lovász numbers. Another way to ask “What is the Lovász number of the plane” would be to adopt the point of view that the Lovász number of a graph has to do with extremizers on the set of positive semidefinite matrices whose (i,j) entry is nonzero only when i and j are adjacent vertices or i=j. So there must be some question one could ask about the space of positive semidefinite symmetric kernels K(x,y) on R^2 x R^2 which are supported on the locus ||x-y||=1 and the diagonal, which question would rightly be called “What is the Lovász number of the plane?” But I’m not sure what it is.
- Having written this, I wonder whether it might be better, rather than thinking about enclosing ellipsoids of a set of points in R^d, just to think of the n points as an nxd matrix X and compute the singular values of X^T X, which would be kind of an “approximating ellipsoid” to the points. Maybe later I’ll think about what that would measure. Or you can!

There is a theta number of the Euclidean space (and of graphs over metric spaces in general), which can be computed analitically. It provides a lower bound for the _measurable_ chromatic number of the plane, which is provably at least as good as any lower bound provided by the theta number of any finite subgraph of the unit-distance graph.

The best upper bound for the independence number of the plane comes from this theta number (see e.g. http://arxiv.org/abs/1501.00168) + some extra constraints. A detailed description of the idea can be found in my PhD thesis (see in particular Chapter 4 and Sec. 4.6f):

Click to access thesis.pdf

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