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.)

Continue reading

From Peter Cameron’s Blog: The symmetric group 3: Automorphisms

Gil Kalai:

Here is, with Peter’s kind permission, a rebloging of Peter’s post on the automorphism group of S_n. Other very nice accounts are by the Secret blogging seminar;  John Baez,; A paper by Howard, Millson, Snowden, and Vakil; and most famously the legendary Chapter 6 (!) from the book by Cameron and Van-Lint (I dont have an electronic version for it).

My TYI 25 question about it arose from Sonia Balagopalan’s lecture in our combinatorics seminar on the 16 vertex triangulation of 4-dimensional projective space. (Here is the link to her paper.)

Originally posted on Peter Cameron's Blog:

No account of the symmetric group can be complete without mentioning the remarkable fact that the symmetric group of degree n (finite or infinite) has an outer automorphism if and only if n=6.

Here are the definitions. An automorphism of a group G is a permutation p of the group which preserves products, that is, (xy)p=(xp)(yp) for all x,y (where, as a card-carrying algebraist, I write the function on the right of its argument). The automorphisms of G themselves form a group, and the inner automorphisms (the conjugation maps x?g-1xg) form a normal subgroup; the factor group is the outer automorphism group of G. Abusing terminology, we say that G has outer automorphisms if the outer automorphism group is not the trivial group, that is, not all automorphisms are inner.

Now the symmetric group S

View original 1,245 more words

Coloring Simple Polytopes and Triangulations


Edge-coloring of simple polytopes

One of the equivalent formulation of the four-color theorem asserts that:

Theorem (4CT) : Every cubic bridgeless planar graph is 3-edge colorable

So we can color the edges by three colors such that every two edges sharing a vertex are colored by different colors.

Abby Thompson asked the following question:

Question: Suppose that G is the graph of a simple d-polytope with n vertices. Suppose also that n is even (this is automatic if d is odd). Can we always properly color the edges of G with d colors?

Vising theorem reminded

Vising’s theorem asserts that a graph with maximum degree D can be edge-colored by D+1 colors. This is one of the most fundamental theorems in graph theory. (One of my ambitions for the blog is to interactively teach the proof based on a guided way toward a proof, based on Diestel’s book, that I tried in a graph theory course some years ago.) Class-one graphs are those graphs with edge chromatic number equal to the maximum degree. Those graphs that required one more color are called class-two graphs.

Moving to triangulations

Thompson asked also a more general question:

Question: Let G be a dual graph of a triangulation of the (d-1)-dimensional sphere. Suppose that G has an even number of vertices.  Is G d-edge colorable?

Grunbaum’s question and counterexample

Branko Grunbaum proposed a beautiful generalization for the 4CT: He conjectured that the dual graph of a triangulation of every two-dimensional manifold is 3-edge colorable. This conjecture was refuted in 2009 by Martin Kochol.

Triangulations in higher dimensions

A third question, even more general, posed by Thompson is: Let G be a dual graph of a triangulation of a (d-1)-dimensional manifold, d ≥ 4. Suppose that G has an even number of vertices.  Is G d-edge colorable?

Hamiltonian cycles

Coloring graph is notoriously difficult but finding a Hamiltonian cycle is even more difficult.

Tait’s conjecture and Barnette’s conjectures

Peter Tait conjectured in 1884 that every 3-connected cubic planar graph is Hamiltonian. His conjecture was disproved by William Tutte in 1946. A cubic Hamiltonian graph must be of class I and therefore Tait’s conjecture implies the 4CT. David Barnette proposed two ways to save Tait’s conjecture: one for adding the condition that all faces have an even number of edges or, equivalently that the graph is bipartite, and another, by moving up in the dimension.

Barnette’s conjecture I: Planar 3-connected cubic bipartite graphs are Hamiltonian.

Barnette’s conjecture II: Graphs of simple d-polytopes d ≥ 4 are Hamiltonian.

Barnette’s hamiltonicity conjecture in high dimension does not imply a positive answer to Thompson’s quaestion. We can still ask for the following common strengthening:  does the graph of a simple d-polytope, d 4, with an even number of vertices contain [d/2] edge-disjoint Hamiltonian cycles?

There are few more things to mention: Peter Tait made also three beautiful conjectures about knots. They were all proved, but it took a century more or less. When we move to high dimensions there are other notions of coloring and other generalizations of “Hamiltonian cycles.” You can Test Your Imagination and try to think about such notions!

Update (Dec 7): Following rupeixu’s comment I asked a question over: generalizations-of-the-four-color-theorem.

Test your intuition 24: Which of the following three groups is trivial


Martin Bridson

We have three finitely presented groups

A is generated by two generators a and b and one relation  a^{-1} \cdot b\cdot a = b^2

B is generated by three generators a, b, c and three relations a^{-1} \cdot b\cdot a = b^2,  b^{-1}\cdot c\cdot b = c^2,  c^{-1}\cdot a\cdot c = a^2.

C is generated by four generators a, b, c, d and four relations a^{-1} \cdot b\cdot a = b^2,  b^{-1}\cdot c\cdot b = c^2,  c^{-1}\cdot d\cdot c = d^2, and d^{-1}\cdot a \cdot d = a^2.

Test your intuition: which of the groups A, B or C is trivial

Please do not answer this poll if you already knew the answer

This poll is to learn how many people already knew the answer before. Please please answer.

As always comments are welcome!

Update: I did not know the answer (and I feel now better about it).

School Starts at HUJI

We are now starting the third week of the academic year at HUJI. As usual, things are very hectic, a lot of activities in the mathematics department, in our sister CS department, around in the campus, and in our combinatorics group. A lot is also happening in other universities around. This semester I am teaching a course on “Social Choice and some Topics from Cooperative Game Theory” in our Federmann Center for the Study of Rationality. I will probably create a page for the course in the near future. During the summer we ran an informal multi-university research activity on analysis of Boolean functions where for two months we met every week for a whole day. I will try to report on what we were studying  and we will probably meet during the academic year 2-3 times each semester.

A few of many future events: Later this month we will have here a cozy Polish-Israeli meeting on  topological combinatorics, on December we will celebrate Ilya Rips 65 birthday with a conference on  geometric and combinatorial group theory, and at the last two weeks of January our traditional The 18th yearly midrasha (school) in mathematics that will be devoted to combinatorics with six lecture series and a few additional talks aimed at teaching some of the very latest exciting developments. If you did not register yet to the Midrasha, please go ahead and do so. This can be a very nice opportunity to visit Israel and learn some exciting combinatorics and to meet people. Partial support for travel and local expenses is available.

COMBSEM – weeks I, II, III

Our weekly combinatorics seminar is meeting on Mondays 9-11. Let me tell you a little on what we had in the first two weeks and what is planned for the third.

Week I:  A startling extension of the associahedron

Monday, October 27, 11:00–13:00, at room B221 in Rothberg building
(new CS and engineering building).

Speaker: Jean-Philippe Labbé, HU

Title: A construction of complete multiassociahedric fans

Originally, Coxeter groups emerged as an algebraic abstraction of
groups generated by reflections in a vector space. The relative
generality of their definition allows them to be related to many
combinatorial, geometric and algebraic objects. This talk show cases
recent developments in the study of a family of simplicial spheres
describing multi-triangulations of convex polygons based on
combinatorial aspects of Coxeter groups of type A. This family of
simplicial spheres generalizes the associahedra. A conjecture of
Jonsson (2003) asserts that these simplicial spheres can be realized
geometrically as the boundary complex of a convex polygon, that would
be thus called multiassociahedron. We will describe a
construction method to obtain complete simplicial fans realizing an
infinite non-trivial family of multi-associahedra.  At it turned out,  Jonsson’s conjecture is closely related to a conjecture by Miller and Knudson.
This is joint work with Nantel Bergeron and Cesar Ceballos (Fields
Institute and York Univ.).

Week II: Finally, progress on Withenhausen’s problem in 2 dimension

Speaker: Evan deCorte, HU

Title: Spherical sets avoiding a prescribed set of angles

Abstract: Let X be any subset of the interval [-1,1]. A subset I of
the unit sphere in R^n will be called X-avoiding if <u,v> is not in X
for any u,v in I. The problem of determining the maximum surface
measure of a {0}-avoiding set was first stated in a 1974 note by H.S.
Witsenhausen; there the upper bound of 1/n times the surface measure
of the sphere is derived from a simple averaging argument. A
consequence of the Frankl-Wilson theorem is that this fraction
decreases exponentially, but until now the 1/3 upper bound for the
case n=3 has not moved. We improve this bound to 0.313 using an
approach inspired by Delsarte’s linear programming bounds for codes,
combined with some combinatorial reasoning. In the second half of the
talk, we turn our attention to the following question: Does there
exist an X-avoiding subset of the unit sphere maximizing the surface
measure among all X-avoiding subsets? (Or could there be a supremum
measure which is never attained as a maximum?) Using a combination of
harmonic and functional analysis, we show that a maximizer must exist
when n is at least 3, regardless of X. When n=2, the existence of a
maximizer depends on X; sometimes it exists, sometimes it does not.
This is joint work with Oleg Pikhurko.

Week III (coming up tommorow): Ramsey numbers for cliques vs cubes conquered at last!

Speaker: Gonzalo Fiz Pontiveros, HU

Title:  The Ramsey number of the clique and the hypercube

The Ramsey number r(K_s,Q_n) is the smallest positive integer N
such that every red-blue colouring of the edges of the complete graph
K_N on N vertices contains either a red n-dimensional hypercube,
or a blue clique on s vertices. It was conjectured in 1983 by
Erdos and Burr that r(K_s,Q_n)=(s-1)(2^{n}-1)+1 for every
positive integer s and every sufficiently large n. The aim of the
talk is to give an overview of the proof of this result and, if time
allows it, discuss some related problems.

Joint work with S. Griffiths, R.Morris, D.Saxton and J.Skokan.

A lecture by Noga


Noga with Uri Feige among various other heroes

A few weeks ago I devoted a post to the 240-summit conference for Péter Frankl, Zoltán Füredi, Ervin Győri and János Pach, and today I will bring you the slides of Noga Alon’s lecture in the meeting. Noga is my genious twin academic brother – we both were graduate students under the supervision of Micha A. Perles in the same years and we both went to MIT as postocs in fall 1983.  The lecture starts with briefly mentioning four results by the birthday boys related to combinatorics and geometry and continues with recent startling results by Alon, Ankur Moitra, and Benny Sudakov. One out of many contributions of Noga over the years is building a large infrastructure of constructions and examples, often very surprising,  in combinatorics, graph theory, information theory,TOC, and related areas. And the new results add to this infrastructure. The slides are very clear. Enjoy!