# When Do a Few Colors Suffice?

When can we properly color the vertices of a graph with a few colors? This is a notoriously difficult problem. Things get a little better if we consider simultaneously a graph together with all its induced subgraphs. Recall that an induced subgraph of a graph G is a subgraph formed by a subset of the vertices of G together with all edges of G spanned  on these vertices.  An induced cycle of length larger than three is called a hole, and an induced subgraph which is a complement of a cycle of length larger than 3 is called an anti-hole. As usual, $\chi (G)$ is the chromatic number of G and $\omega (G)$ is the clique number of G (the maximum number of vertices that form a complete subgraph. Clearly, for every graph G

$\chi(G) \ge \omega (G)$.

## Perfect graphs

Question 1: Describe the class $\cal G$  of graphs closed under induced subgraphs, with the property that $\chi(G)=\omega (G)$ for every $G\in{\cal G}$.

A graph G is called perfect if  $\chi(H)=\omega (H)$ for every induced subgraph H of G. So Question 1 asks for a description of perfect graphs. The study of perfect graphs is among the most important areas of graph theory, and much progress was made along the years.

Interval graphs, chordal graphs, comparability graphs of POSETS  , … are perfect.

Two major theorems about perfect graphs, both conjectured by Claude Berge are:

The perfect graph theorem (Lovasz, 1972): The complement of a perfect graph is perfect

The strong perfect graph theorem (Chudnovsky, Robertson, Seymour and Thomas, 2002): A graph is perfect if and only if it does not contain  an odd hole and an odd anti-hole.

## Mycielski Graphs

There are triangle-free graphs with arbitrary large chromatic numbers. An important construction by Mycielski goes as follows: Given a triangle graph G with n vertices $v_1,v_2, \dots, v_n$ create a new graph  G’ as follows: add n new vertices $u_1, u_2\dots u_n$ and a vertex w. Now add w to each $u_i$ and for every i and j for which $v_i$ and $v_j$ are adjacent add also an edge between $v_i$ and $u_j$ (and thus also between $u_i$ and $v_j$.)

## Classes of Graphs with bounded chromatic numbers

Question 2: Describe classes of graphs closed under induced subgraphs with bounded chromatic numbers.

Here are three theorems in this direction. The first answers affirmatively a conjecture by Kalai and Meshulam. The second and third prove conjectures by Gyarfas.

## Trinity Graphs

The Trinity graph theorem (Bonamy, Charbit and Thomasse, 2013): Graphs without induced cycles of  length divisible by three have bounded chromatic numbers.

(The paper: Graphs with large chromatic number induce 3k-cycles.)

### Steps toward Gyarfas conjecture

Theorem (Scott and Seymour, 2014):  Triangle-free graphs without odd induced cycles have bounded chromatic number.

(The paper:  Coloring graphs with no odd holes.)

# Around Borsuk’s Conjecture 3: How to Save Borsuk’s conjecture

Borsuk asked in 1933 if every bounded set K of diameter 1 in $R^d$ can be covered by d+1 sets of smaller diameter. A positive answer was referred to as the “Borsuk Conjecture,” and it was disproved by Jeff Kahn and me in 1993. Many interesting open problems remain.  The first two posts in the series “Around Borsuk’s Conjecture” are here and here. See also these posts (I,II,III, IV), and the post “Surprises in mathematics and theory” on Lipton and Reagan’s blog GLL.

Can we save the conjecture? We can certainly try, and in this post I would like to examine the possibility that Borsuk’s conjecture is correct except from some “coincidental” sets. The question is how to properly define “coincidental.”

Let K be a set of points in $R^d$ and let A be a set of pairs of points in K. We say that the pair (K, A) is general if for every continuous deformation of the distances on A there is a deformation K’ of K which realizes the deformed distances.

(This condition is related to the “strong Arnold property” (aka “transversality”) in the theory of Colin de Verdière invariants of graphs; see also this paper  by van der Holst, Lovasz and Schrijver.)

Conjecture 1: If D is the set of diameters in K and (K,D) is general then K can be partitioned into d+1 sets of smaller diameter.

We propose also (somewhat stronger) that this conjecture holds even when “continuous deformation” is replaced with “infinitesimal deformation”.

The finite case is of special interest:

A graph embedded in $R^d$ is stress-free if we cannot assign non-trivial weights to the edges so that the weighted sum of the edges containing any  vertex v (regarded as vectors from v) is zero for every vertex v. (Here we embed the vertices and regard the edges as straight line segments. (Edges may intersect.) Such a graph is called a “geometric graph”.) When we restrict Conjecture 1 to finite configurations of points we get.

Conjecture 2: If G is a stress free geometric graph of diameters in $R^d$  then G is (d+1)-colorable.

A geometric graph of diameters is a geometric graph with all edges having the same length and all non edged having smaller lengths. The attempt for “saving” the Borsuk Conjecture presented here and Conjectures 1 and 2 first appeared in a 2002 collection of open problems dedicated to Daniel J. Kleitman, edited by Douglas West.

When we consider finite configurations of points  we can make a similar conjecture for the minimal distances:

Conjecture 3: If the geometric graph of pairs of vertices realizing the minimal distances of a point-configuration in $R^d$ is stress-free, then it is (d+1)-colorable.

We can speculate that even the following stronger conjectures are true:

Conjecture 4: If G is a stress-free geometric graph in $R^d$ so that all edges in G are longer than all non-edges of G, then G is (d+1)-colorable.

Conjecture 5: If G is a stress-free geometric graph in $R^d$ so that all edges in G are shorter than all non-edges of G, then G is (d+1)-colorable.

We can even try to extend the condition further so edges in the geometric graph will be larger (or smaller) than non-edges only just “locally” for neighbors of each given vertex.

1) It is not true that every stress-free geometric graph in $R^d$ is (d+1)-colorable, and not even that every stress-free unit-distance graph is (d+1)-colorable. Here is the (well-known) example referred to as the Moser Spindle. Finding conditions under which stress-free graphs in $R^d$ are (d+1)-colorable is an interesting challenge.

2) Since a stress-free graph with n vertices has at most $dn - {{d+1} \choose {2}}$ edges it must have a vertex of degree 2d-1 or less and hence it is 2d colorable. I expect this to be best possible but I am not sure about it. This shows that our “saved” version of Borsuk’s conjecture is of very different nature from the original one. For graphs of diameters in $R^d$ the chromatic number can, by the work of Jeff and me be exponential in $\sqrt d$.

3) It would be interesting to show that conjecture 1 holds in the non-discrete case when  d+1 is replaced by 2d.

4) Coloring vertices of geometric graphs where the edged correspond to the minimal distance is related also the the well known Erdos-Faber-Lovasz conjecture..

See also this 1994 article by Jeff Kahn on Hypergraphs matching, covering and coloring problems.

5) The most famous conjecture regarding coloring of graphs is, of course, the four-color conjecture asserting that every planar graph is 4-colorable that was proved by Appel and Haken in 1976.  Thinking about the four-color conjecture is always both fascinating and frustrating. An embedding for maximal planar graphs as vertices of a convex 3-dimensional polytope is stress-free (and so is, therefore, also a generic embedding), but we know that this property alone does not suffice for 4-colorability. Finding further conditions for  stress-free graphs in $R^d$ that guarantee (d+1)-colorability can be relevant to the 4CT.

An old conjecture of mine asserts that

Conjecture 6: Let G be a graph obtained from the graph of a d-polytope P by triangulating each (non-triangular) face with non-intersecting diagonals. If G is stress-free (in which case the polytope P is called “elementary”) then G is (d+1)-colorable.

Closer to the conjectures of this post we can ask:

Conjecture 7: If G is a stress-free geometric graph in $R^d$ so that for every edge  e of G  is tangent to the unit ball and every non edge of G intersect the interior of the unit ball, then G is (d+1)-colorable.

### A question that I forgot to include in part I.

What is the minimum diameter $d_n$ such that the unit ball in $R^n$ can be covered by n+1 sets of smaller diameter? It is known that $2-C'\log n/n \le d_n\le 2-C/n$ for some constants C and C’.