1912-2015: 100+ years to Chromatic Polynomial Outline: Origins of the connection-contraction The roots of P(G, λ) The coefficients of P(G, λ ) The roots

1912-2015: 100+ years to Chromatic

Polynomial

Outline:• Origins of the connection-contraction• The roots of P(G,λ)• The coefficients of P(G,λ)• The roots of P(G,λ) for planar G• The all-integer roots of P(G,λ) (for any G)• Tutte Polynomial• Potts Model• Chromatic Polynomial of a Mixed hypergraph• WHO KNEW! (application in cyber security)

04/18/23 People + Ideas = History 2

Mathematical commercial:Mathematical commercial:

04/18/2304/18/23 33People + Ideas = HistoryPeople + Ideas = History

Let us imagine year 1912...

• Just a few decades passed since Gauss published his second proof of the fundamental theorem of algebra, saying that every polynomial of degree n with complex coefficients has precisely n roots.

• At that time it appeared that the theory of polynomials is so powerful and universal that it can solve almost any problem.


George David Birkhoff: 1884-1944


• 1912: working at Princeton, published the paper: "A determinant formula for the number of ways of coloring a map", Ann. of Math. 14, 42 –- 46. • the main goal – proof of the four color problem

Career • Birkhoff obtained undergraduate degree from

Harvard. • He completed his Ph.D. in 1907, on differential

equations, at the University of Chicago. • While Moore was his supervisor, he was most

influenced by the writings of Henri Poincaré. • After teaching at the University of Wisconsin and

Princeton University, he taught at Harvard University from 1912 until his death.


George David Birkhoff: 1884-1944


• 1935: Menzel, Einstein and Birkhoff,

• the time of Einstein's receiving an honorary degree from Harvard; Cambridge, Massachusetts;

•Date: Unknown

Career • Birkhoff worked on many different

mathematical topics. His main work was on dynamics and ergodic theory . His ergodic theorem (1932) transformed the Maxwell - Boltzmann kinetic theory of gases into a rigorous principle.

• This theory, which resolved one of the fundamental problems arising in the theory of gases and statistical mechanics, has been influential not only in dynamics itself but also in probability theory, group theory, and functional analysis.


Birkhoff’s 10 doctoral students:• David Bourgin

Robert D. CarmichaelHyman EttlingerBernard KoopmanRudolph LangerMarston MorseMarshall H. StoneJoseph L. WalshHassler Whitney (1932 Dissertation: The Coloring of Graphs )David Widder



1912:


1912:

The “plan”:• The main goal of the paper was trying to prove

the famous four color problem by algebraic method.

• The chromatic polynomial of a graph (of a map at that time!) is a polynomial P(λ) which gives the number of proper colorings using at most λ colors.

• The four color problem is equivalent to ”simply” answering the question: is it true that P(4)> 0 for any planar graph?


WHO KNEW!• Though the main goal was never achieved by this method,

the concept of chromatic polynomial generated an entire field of research with many new ideas, concepts, methods, discrete structures and generalizations

• Applications: as a tool of combinatorics with a range from computer science to statistical mechanics.

• The concept of chromatic polynomial is so fundamental that many areas of unforeseen applications appeared much later than 1912.

• Currently Data Base of the American Mathematical Society contains abstracts of 830 refereed papers on chromatic polynomials (8.14 papers/year).


Hassler Whitney: 1907-1989


1932: published the paper “The colorings of graphs” (Ann. Of Math. 33 (1932) 688-718

Hassler Whitney: 1907-1989


Hassler Whitney, 1986

Whitney's earliest work, from 1930 to 1933, was on graph theory under supervision of Birkhoff.

Many of his results were in graph coloring, and the final proof (1977) of the four-color problem in part relied on his results.

His work in graph theory culminated in a 1935 paper, where he laid the foundations for matroids, a fundamental notion in modern Combinatorics and representation theory.


1932:

Next slide: Embryo of “connection-contraction”:

• Note added in the proof by Forster:

• G’=G-{a,b} = removing of edge {a,b}

• G”= “coalesce vertices a and b”



1932:

Alexander Zykov:• 1949 published paper “On some properties of

linear complexes”

• Next slide: the first explicit description (in drawings) of the connection-contraction algorithm for calculation of the chromatic polynomial of any graph



1949:

Alexander Zykov (1922-2013)


Odessa, Ukraine, 2009

With Zykov and Vizing 9/11/2001


Connection-contraction algorithm:


= +

G G1G2

contractionconnectionnon adjacent

P(G, λ) = P(G1, λ) + P(G2, λ) =… =at the very end, we obtain a combination of

complete graphs and their chromatic polynomials

a b a b ab

Deletion-contraction algorithm:


= -G G1

G2

contractiondeletion adjacent

P(G, λ) = P(G1, λ) - P(G2, λ) =… = at the very end, we obtain a combination of

empty graphs and their chromatic polynomials

a b a b ab

Zykov ß-polynomial (1976):


= + ß

G G1 G2

contractionconnectionnon adjacent

H(G, ß) =1 +a1ß+a2ß2 +… = combinations of complete graphs…

a b a b ab

Zykov ß-polynomial (1976)


x n

nK

nH

1nH

nn HDH ]1[ 21

Differentiation operator for ß-polynomial

(never published)

Add simplicial vertex of degree n-ν

Coefficients of P(G,λ)


Coefficients of P(G,λ): Birkhoff (1912): the chromatic polynomial of a

graph has the form

where is the number of feasible partitions of vertex set into classes and

the falling factorial


n

i

ii GrGP

1

)()(),(

)(Grii

)1)...(2)(1()( ii

Coefficients of P(G,λ):• Whitney, 1932, the broken circuit theorem: • if then


in

iiaGP

1

),(

||

0

),()1(E

r

ri riSpa

where Sp(i,r) is the number of spanning subgraphs of G having i components and r edges (embryo of the concept of matroid -1935).

Coefficients of P(G,λ):• The degree of P(G,λ) is n=|V|, the number of

vertices • The leading coefficient is always 1• The coefficient with λn-1 is -|E|• Constant term is always 0• The coefficients alternate in sign


Coefficients of P(G,λ):

• Read (1968) Unimodal Conjecture: There exists such that

• It is true for several classes of graphs. • There is information that it is proved (not published)


k

||||...||...|||| 121 nnk aaaaa

The roots of P(G,λ) for any G



• Integers 0,1,2,…χ(G)-1 are always the roots because by definition of the chromatic number χ(G), there are no colorings with any of these numbers of colors

• Therefore we are next interested in roots other than set {0,1,2,…, χ(G)-1}.

• Chromatic polynomial has no real root greater than n-1.


The roots of P(G,λ) for any G• Intervals (-∞,0) and (0,1) are root-free for all

graphs• Jackson (1993) proved interval (1,32/27] is also a

root-free interval• Thomassen (1997) proved that for any interval

(a, b) with 32/27≤ a<b, there exists a graph having a chromatic root in this interval

• -∞

04/18/23 People + Ideas = History 34Dense rootsNO roots

10

27

32


• Read and Tutte (1988): P(G, τ+1)≠0 for any graph G where

τ=(1+√5)/2 = …golden ratio (1.618)

• Irrationals whose square is rational are never the roots too…



• Sokal (2001): There exists a universal constant M such that if G has maximum degree Δ, then all complex roots of P(G,λ) satisfy |z| < M Δ.

• Sokal (2004): The complex roots of all chromatic polynomials are dense in complex plane.


The roots of P(G,λ) for planar G



• Birkhoff and Lewis (1946): for any planar graph G, P(G,λ)>0 for all real λ≥ 5;

[5,∞) is root-free • Appel and Haken, the Four Color Theorem,

1977: for any planar graph G, P(G,4)>0.• Birkhoff and Lewis (1946): CONJECTURE: for planar graphs, (4,5) is root-free



• Tutte has shown that for planar graphs P(G,τ+ 2) > 0 where τ is the golden ratio; Since τ+ 2≈ 3.6183 it was close to … 4.• Royle (2001): there are planar graphs with the

chromatic roots arbitrarily close to 4 from the left...

• The Four Color Problem has slipped away… again!



• Thomassen (1997): Planar graphs have chromatic roots arbitrarily close to 32/27 from the right

CONJECTURE: the set of chromatic roots of planar graphs consists of 0, 1 and a dense subset of interval (32/27,4)


540 1NO roots

?

NO roots32/27 2 3 ?

? ? ?

The all-integer roots of P(G,λ) (for any G)




It is evident that every chromatic polynomial has the roots 0,1,2,…,χ-1 (perfect set of roots).

Question: when these numbers are the ONLY roots of the chromatic polynomial?

No one had any idea for so long time… As sometimes happens, the answer came decades later from “NOWHERE”…


• 1953: Watson and Crick have discovered the linear structure of DNA molecule. Gene= interval of DNA molecule.

• Hajnal and Surányi (1958) : introduced interval graphs and proved they are chordal (every cycle of length >=4 has a chord)

• Dirac (1961): a graph is chordal iff every minimal separator is a clique

• every chordal graph has two simplicial vertices… 04/18/23 People + Ideas = History 43

The integer roots of P(G,λ) (for any G)


mK

xG

G

),()(),( xGPmGP

x

Simplicial vertex of degree m

The root is the degree of a simplicial vertex when we delete it

m

The all-integer roots of P(G,λ)


Klaus, Kretz, Walter, Walter (1974) have rediscovered chordal graphs and proved (without using simplicial vertices!)

Theorem: If G is a chordal graph, then:

The converse (conjectured) is not true:

Read (1974): counterexample is if we put one vertex on any edge. Such graph is not chordal and

1210 )1...()2()1(),( ssssGP

6K

)4()3)(2)(1(),( 3 GP

Tutte Polynomial


William Tutte (1917-2002)


19991936

Tutte polynomial (1954)• Tutte defined two-variable polynomial as a generalization of

the chromatic polynomial and the deletion-contraction algorithm;

• Many graph polynomials coming from different areas of mathematics and even physics (like Flow polynomial, Reliability polynomial, Jones polynomial of alternating knots, Partition function of the Potts model) are in fact special cases of the Tutte polynomial;

• It is also the most general graph invariant that can be defined by deletion–contraction algorithm.


Tutte polynomial:

),,( yxGT


)( eGxT

)( eGyT

)()( eGTeGT

if is a bridgee

if is a loope

otherwise

1 if nEG

Tutte polynomial:• For a general undirected graph


:),( EVG

||||)()()( )1()1(),,( VAAc

EA

EcAc yxyxGT

Where is the number of connected components of graph

)(Ac

),( AV

Tutte polynomial:• For any tree with m edges T(G, x,y)= xm

• For every forest G with m edges and k loops T(G, x, y)= xm yk

• For a planar graph G and its dual G* T(G, x, y) = T(G*, y, x)


Tutte polynomial:• T(1, 1) = number of spanning trees of G• T(1, 2) = number of connected spanning subgraphs• T(2, 1) = number of acyclic subgraphs (forests)• T(2, 2) = number of spanning subgraphs• T(2, 0) = number of acyclic orientations of G • T(1, 0) = number of acyclic orientations where the only

source is a fixed vertex• T(0, 2) = number of orientations of a bridgeless G such

that each edge is contained in an oriented cycle• T(-2; 0) =the number of Eulerian orientations• T(-1,-1) =the dimension of a space of binary codes


Tutte Polynomial contains the Chromatic Polynomial:

),( GP


)0,1,()1( )()(|| GTGcGcV

Potts model


Potts model (1952):• Ferromagnets = set of interacting spins on a

crystalline lattice• Each spin can assume one of q possible states• If two neighboring spins are in the same state, it

adds some value to the energy of the system• The Boltzmann weight of a spin configuration

(=coloring) is where and is the temperature• The probability of spin configuration is

proportional to Boltzmann weight.


He 01

kT

T

Potts model• The sum of Boltzmann weights of all

configurations is the partition function in the Potts model

• It is a polynomial in two variables Z(G, )• The behavior of the system is determined by the

possibility of (adjacent) spins to get the same value

• When temperature function Z becomes the chromatic polynomial P(G, q) of the crystalline lattice (ferromagnet) G


,q

0T

Potts model:• Phase transitions, of particular

importance for statistical physicists, are closely related to the roots of the partition function, and therefore to the roots of the chromatic polynomial.


Tutte Plane for


Flow polynomial:-axis

Chromatic polynomial: -axis

Spanning subgraphs

Spanning trees

Eulerian orientations

Ising ferromagnetic =Potts partition function along (x-1)(y-1)=q

Jones polynomial:xy=1

Acyclic orientationsAcyclic orientations with a single fixed source

Connected spanning subgraphs

),,( yxGT

x

y

)1,()1()0,2,( || GPGT V

Chromatic polynomial of Mixed Hypergraphs


The next turn: mixed hypergraph coloring (1993):


Mixed hypergraph:

Proper coloring of vertices using ≤ λ colors:

Every C-edge has 2 vertices of Common color, and Every D-edge has 2 vertices of Different colors

Connection-contraction algorithm from graphs

generalizes to the Splitting-Contraction Algorithm for mixed hypergraphs

),,( DCXH

Connection-contraction becomes Splitting-Contraction:


=

Two vertices not connected by edge of size 2

combinations of complete graphs…

+

C-edges:

D-edges:

...),(),(),( 21 HPHPHP

Common color Distinct colors

),,( DCXH ),,( 12 DCXH ),,( 11 DCXH

New properties of P(H,λ)• If H is uncolorable, then P(H,λ)=0 (never

happened!)• The degree of P(H,λ) is the upper chromatic

number χ’(H) (not n as in graphs; just coincidence!)• The leading coefficient is (not 1 as in graphs,

just another special case!)• Generally:


)(' Hr

'

)()(),(

i

ii HrHP

It was n since Birkhoff

New properties of P(H,λ)• Mixed hypergraph H may have PHANTOM

(invisible) VERTICES (never happened!):

• The chromatic spectrum may be broken (never happened):

•


),(),( xHPHP

)0,...,0,,...0,...,,0,...,0()( ' rrHR

First positive component Last positive component

GAP

• Coalescence of regions (Birkhoff)• Contractions of edges (Whitney, Zykov, Tutte)• Was a special case of C-edges• C-edge (not polychromatic subset)• Tip of the iceberg! • Mixed hypergraph: an interaction between

DIFFERENCE AND IDENTITY as philosophical categories


Philosophy behind the concept

WHO KNEW!


The newest application: Byzantine agreement and cyber-security


LESLIE LAMPORT (LaTex author), ROBERT SHOSTAK, and MARSHALL PEASE

(1982): The Byzantine Generals Problem

Reliable computer systems must handle malfunctioning components that give conflicting information to different parts of the system. This situation can be expressed abstractly in terms of a group of generals of the Byzantine army camped with their troops around an enemy city.

Communicating only by messenger, the generals must agree upon a common battle plan (to attack or retreat). However, one or more of them may be traitors (betrayers) who will try to confuse the others.



The problem is to find an algorithm to ensure that the loyal generals will reach agreement. It is shown that, using only oral messages, this problem is solvable if and only if more than two-thirds of the generals are loyal.

So a single traitor can confuse at most two loyal generals. Within these constraints, the problem is solvable for any number of generals and possible traitors.


• Jaffe (UW), Mascibroda (Microsoft), Sen (Princeton): On the Price of Equivocation in Byzantine Agreement (2012)

Processors(computers) = vertices , Partial Broadcast Channels = hyperedges

In Byzantine agreement problem, a set of n processors, any f of whom may be arbitrarily faulty, must reach agreement on a value proposed by one correct (main) processor

Faulty processors = under control of malicious adversary04/18/23 People + Ideas = History 68



Equivocation is fundamentally an act involving three parties: a faulty processor that lies to two correct processors.

“We model a system of n processors as a 3-uniform, n-vertex hypergraph H = (V,E) where each edge represents a partial broadcast channel. For a fixed integer f, we analyze the conditions under which Byzantine agreement is possible in H, when up to f processors are faulty”.



The authors introduce the concept of h-disjointness which can be seen as “a generalization of a rich body of work on mixed hypergraph coloring and the upper chromatic number (see Voloshin’s book [42]).”

WHO KNEW!

Omitting many details , long story –short:



A k-heterochromatic coloring of a hypergraph H a k-coloring of vertices such that at least one edge is polychromatic.

“In Byzantine agreement problem , a primary line of research is to analyze f (n,k), the minimum number of edges among k-heterochromatically colorable, k-uniform, n-vertex hypergraphs.”

If a hypergraph has less than f(n,k) edges, then in k-coloring every edge has at least two vertices of the same color, what means we deal with a C-hypergraph coloring, the special case of mixed hypergraph coloring.



This breakthrough paper was recently reported at ACM Symposium on Distributed Computing and has received the Best Student Paper Award at Computer Science Department in Princeton University (one of the authors is Google PhD fellow at Princeton).

The paper itself:


PhD Thesis at Princeton


PhD Thesis at Princeton


WHO KNEW??

•PAUL ERDÖS KNEW IT!!!



Marceille 1995


WHO KNEW?!


The very first question was …


Herbert S. Wilf: 1931-2012

One of the papers was …


CONCLUSION


Mixed Hypergraph Coloring today


Possible problems for research:

• Differentiation of chromatic plynomials• The eigenvalues of self dual hypergraphs• Mixed Ramsey Hypergraphs - spectrum


My personal thanks to Birkhoff

• Thanks to George David Birkhoff for his ingenious idea of the chromatic polynomial and everybody mentioned in this talk who explicitly and implicitly helped me in my math career

• If there was no Birkhoff with his paper 100 years ago, we would have a different talk today.


What a world!!!



THANK YOU!For contribution to the theory of chromatic polynomial by inviting me to give this talk and actually attending it!

Memories…


Final thanks to …


References• Originals of many papers • Data Base of the AMS MathSciNet• Dong, Koh, Teo. Chromatic Polynomials and

Chromaticity of Graphs. World Scientific, 2005• Voloshin. Coloring Mixed Hypergraphs (2002) • Internet: Google search, WolframMathWorld,

Wikipedia, etc.


Documents

1912-2015: 100+ years to Chromatic Polynomial Outline: Origins of the connection-contraction The roots of P(G, λ) The coefficients of P(G, λ ) The roots