## Metric chromatic numbers and Lovasz numbers

In the first post of this series I asked whether there was a way to see the Lovasz number of a graph as a chromatic number.  Yes!  I learned it from these notes, written by Big L himself.

Let M be a metric space, and let’s assume for simplicity that M has a transitive group of isometries.  Now write r_M(n) for the radius of the smallest ball containing n points whose pairwise distances are all at least 1.  (So this function is controlling how sphere-packing works in M.)

Let Γ be a graph.  By an M-coloring of Γ we now mean a map from v(Γ) to M such that adjacent vertices are at distance at least 1.  Write χ_Γ(M) for the radius of the smallest disc containing an M-coloring of Γ.  Then we can think of r^{-1}(χ_Γ(M)) as a kind of “M-chromatic number of Γ.”  Scare quotes are because r isn’t necessarily going to be an analytic function or anything; if I wanted to say something literally correct I guess I would say the smallest integer n such that r_M(n) >= χ_Γ(M).

The M-chromatic number is less than the usual chromatic number χ_Γ;  more precisely,

χ_Γ(M) <= r_M(χ_Γ)

Easy:  if there’s an n-coloring of Γ, just compose it with the map from [n] to M of radius r_M(n).  Similary, if ω_Γ is the clique number of Γ, we have

r_M(ω_Γ) <= χ_Γ(M)

because a k-clique can’t be embedded in a ball of radius smaller than r_M(k).

So this M-chromatic number gives a lower bound for the chromatic number and an upper bound for the clique number, just as the Lovasz number does, and just as the fractional chromatic number does.

Example 1:  Lovasz number.  Let M be the sphere in infinite-dimensional Euclidean space.  (Or |Γ|-dimensional Euclidean space, doesn’t matter.)  For our metric use (1/sqrt(2)) Euclidean distance, so that orthogonal vectors are at distance 1 from each other.  If n points are required at pairwise distance at least 1, the closest way to pack them is to make them orthonormal (I didn’t check this, surely easy) and in this case they sit in a ball of radius 1-sqrt(1/2n) around their center of mass.  So r_M(n) = 1 – sqrt(1/2n).  Define t(Γ) to be the real number such that

$1 - \sqrt{1/2t(\Gamma)} = \chi_\Gamma(M)$.

Now I was going to say that t(Γ) is the Lovasz theta number of Γ, but that’s not exactly the definition; that would be the definition if I required the embedding to send adjacent vertices to points at distance exactly 1.  The answer to this MO question suggests that an example of Schrijver might actually separate these invariants, but I haven’t checked.

So I guess let’s say t(Γ) is a “Lovasz-like number” which is between the clique number and the chromatic number.  And like the Lovasz number, but unlike clique and chromatic numbers, it’s super-easy to compute.  An embedding of v(Γ) in the sphere, up to rotation, is specified by the pairwise distance matrix, which can be an arbitrary postive definite symmetric nxn matrix A with 1’s on the diagonal.  Each edge of Γ now gives an inequality a_{ij} > 1.  When you’re optimizing over a space cut out by linear inequalities in the space of psd matrices, you’re just doing semidefinite programming.  (I am punting a little about how to optimize “radius” but hopefully maximum distance of any vector from center of mass is good enough?)

(Note:  you know what, I’ll bet you can take an embedding like this, subtract a small multiple of the center of mass from all the vectors, and get an embedding of v(Γ) in n-space with all angles between adjacent vectors slightly obtuse, and probably this ends up being exactly the same thing as the vector chromatic number defined in the paper I linked to earlier.)

Where is example 2?  It was supposed to be about the fractional chromatic number but then I realized the way I was setting this up wasn’t correct.  The idea is to let M_b be the space of infinite bit strings with exactly b 1’s and use (1/2b) Hamming distance, so that the distance-1 requirement becomes a requirement that two b-element subsets be disjoint.  But I don’t think this quite fits into the framework I adopted at the top of the post.  I’ll circle back to this if I end up having what to say.

## Coloring graphs with polynomials

More chromatic hoonja-doonja!

Suppose you have a bunch of tokens of different colors and weights.  X_1 colors of weight 1 tokens, X_2 colors of weight 2 tokens, etc.

Let Γ be a graph.  A (weighted) b-coloring of Γ is an assignment to each vertex of a set of tokens with total weight b, such that adjacent vertices have no tokens in common.  Let χ_Γ(X_1, … X_b) be the number of b-colorings of Γ.  I made up this definition but I assume it’s in the literature somewhere.

First of all, χ_Γ(X_1, … X_b) is a polynomial.

Is this multivariable “chromatic polynomial” of any interest?  Well, here’s one place it comes up.  By a degree-b polynomial coloring of Γ we mean an assignment of a monic squarefree degree d polynomial in R[x] to each vertex of Γ, so that adjacent vertices are labeled with coprime polynomials.   Now let U_b(Γ) be the manifold parametrizing degree-b colorings of Γ.

Then the Euler characteristic of U_b(Γ) is χ_Γ(-1,1,0,…0).

I worked this out via the same kind of Lefschetz computation as in the previous post, but once you get the answer, you can actually derive this as a corollary of Stanley’s theorem.  And it was presumably already known.

By the way:  let V_n be the free vector space spanned by the b-colorings of Γ where all the tokens have weight 1; these are called fractional colorings sometimes.  Then S_n acts on V_n by permutation of colors.  The character of this action is a class function on S_n.  More precisely, it is

χ_Γ(X_1, … X_b)

where X_i is now interpreted as a class function on S_n, sending a permutation to the number of i-cycles in its cycle decomposition.  Of course the real thing going on behind the scenes is that the V_n form a finitely generated FI-module.

## Counting acyclic orientations with topology

Still thinking about chromatic polynomials.   Recall: if Γ is a graph, the chromatic polynomial χ_Γ(n) is the number of ways to color the vertices of Γ in which no two adjacent vertices have the same color.

Fact:  χ_Γ(-1) is the number of acyclic orientations of Γ.

This is a theorem of Richard Stanley from 1973.

Here’s a sketch of a weird proof of that fact, which I think can be made into an actual weird proof.  Let U be the hyperplane complement

$\mathbf{A}^|\Gamma| - \bigcup_{ij \in e(\Gamma)} (z_i = z_j)$

Note that |U(F_q)| is just the number of colorings of Γ by elements of F_q; that is,  χ_Γ(q).  More importantly, the Poincare polynomial of the manifold U(C) is (up to powers of -1 and t) χ_Γ(-1/t).  The reason |U(F_q)| is  χ_Γ(q) is that Frobenius acts on H^i(U) by q^{-i}.  (OK, I switched to etale cohomology but for hyperplane complements everything’s fine.)  So what should  χ_Γ(-1) mean?  Well, the Lefschetz trace formula suggests you look for an operator on U(C) which acts as -1 on the H^1, whence as (-1)^i on the H^i.  Hey, I can think of one — complex conjugation!  Call that c.

Then Lefchetz says χ_Γ(-1) should be the number of fixed points of c, perhaps counted with some index.  But careful — the fixed point locus of c isn’t a bunch of isolated points, as it would be for a generic diffeo; it’s U(R), which has positive dimension!  But that’s OK; in cases like this we can just replace cardinality with Euler characteristic.  (This is the part that’s folkloric and sketchy.)  So

χ(U(R)) = χ_Γ(-1)

at least up to sign.  But U(R) is just a real hyperplane complement, which means all its components are contractible, so the Euler characteristic is just the number of components.  What’s more:  if (x_1, … x_|Γ|) is a point of U(R), then x_i – x_j is nonzero for every edge ij; that means that the sign of x_i – x_j is constant on every component of U(R).  That sign is equivalent to an orientation of the edge!  And this orientation is obviously acyclic.  Furthermore, every acyclic orientation can evidently be realized by a point of U(R).

To sum up:  acyclic orientations are in bijection with the connected components of U(R), which by Lefschetz are χ_Γ(-1) in number.

## Metric chromatic numbers

Idle thought.  Let G be a (loopless) graph, let M be a metric space, and let b be a nonnegative real number.  Then let Conf(G,M,b) be the space of maps from the vertices of the graph to M such that no two adjacent vertices are within b of each other.

When b=0 and G is the complete graph K_n, this is the usual ordered configuration space of n points on M.  When G is the empty graph on n vertices, it’s just M^n.  When M is a set of N points with the discrete metric, Conf(G,M,b) is the same thing for every b;  a set of points whose cardinality is χ_G(N), the chromatic polynomial of G evaluated at N.  When M is a box, Conf(G,M,b) is the “discs in a box” space I blogged about here.  If M is (Z/2Z)^k with Hamming distance, you are asking about how many ways you can supply G with k 2-colorings so that no edge is monochromatic in more than k-b-1 of them.

Update:  Ian Agol links in the comments to this paper about Conf(G,M,0) by Eastwood and Huggett; the paper points out that the Euler characteristic of Conf(G,M,0) computes χ_G(N) whenever M has Euler characteristic N; so M being N points works, but so does M = CP^{N-1}, and that’s the case they study.

When b=0 and G is the complex plane, Conf(G,C,0) is the complement of the graphic arrangement of G; its Poincare polynomial is  t^-{|G|} χ_G(-1/t).  Lots of graphs have the same chromatic polynomial (e.g. all trees do) but do they have homeomorphic Conf(G,C,0)?

This is fun to think about!  Is Conf(G,M,0), for various manifolds other than discrete sets of points, an interesting invariant of a graph?

You can think of

vol(Conf(G,M,b)) vol(M)^{-n}

as a sort of analogue of the chromatic polynomial, especially when b is small; when M = C, for instance, Conf(G,M,b) is just the complement of tubular neighborhoods around the hyperplanes in the graphical arrangement, and its volume should be roughly b^|G|χ_G(1/b) I think.  When b gets big, this function deviates from the chromatic polynomial, and in particular you can ask when it hits 0.

In other words:  you could define an M-chromatic number χ(G,M) to be the smallest B such that Conf(G,M,1/B) is nonempty.  When M is a circle S^1 with circumference 1, you can check that χ(G,M) is at least the clique number of G and at most the chromatic number.  If G is a (2n+1)-cycle, the clique number is 2, the chromatic number is 3, and the S^1-chromatic number is 2+1/n, if I did this right.  Does it have anything to do with the Lovasz number, which is also wedged between clique number and chromatic number?  Relevant here:  the vector chromatic number, which is determined by χ(G,S^{v(G)-1}), and which is in fact a lower bound for the Lovasz number.