chromatic polynomials

8/13/2019 Chromatic Polynomials

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BACKGROUND POLYNOMIALS COMPLEX Z EROS

AN INTRODUCTION TOCHROMATIC

POLYNOMIALS

Gordon Royle

School of Mathematics & StatisticsUniversity of Western Australia

Junior Mathematics Seminar, UWASeptember 2011

GORDONR OYLE CHROMATIC P OLYNOMIALS


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BACKGROUND POLYNOMIALS COMPLEX Z EROS

OUTLINE

1 BACKGROUNDBasics

Graph Colouring

4 Colours Suffice

2 POLYNOMIALS

Chromatic Function

Chromatic Roots

Potts Model

3 COMPLEX ZEROS

Absolute Bounds

Parameterized Bounds



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BACKGROUND POLYNOMIALS COMPLEX Z EROS BASICS GRAPH C OLOURING 4 COLOURS S UFFICE

OUTLINE

1 BACKGROUNDBasics

Graph Colouring

4 Colours Suffice

2 POLYNOMIALS

Chromatic Function

Chromatic Roots

Potts Model

3 COMPLEX ZEROS

Absolute Bounds




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GRAPHS



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GRAPHS

Vertices


C G C 4 C S


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GRAPHS

Vertices

Edges


BAC G O PO O A S CO Z OS BAS CS G A C O O G 4 CO O S S C


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BASIC TERMINOLOGY

Petersen Graph

Phas 10vertices

Phas 15edges

Each vertex has 3 neighbours

Pis3-regular(akacubic)




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POSITION IS IRRELEVANT




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The Petersen graph isnot planar




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The Petersen graph isnot planarandnot hamiltonian




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OUTLINE

1 BACKGROUNDBasics

Graph Colouring

4 Colours Suffice

2 POLYNOMIALSChromatic Function

Chromatic Roots

Potts Model

3 COMPLEX ZEROS

Absolute Bounds





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COLOURING

Aproper colouringof a graph is an assignment ofcoloursto thevertices such thatno edgeis monochromatic.

A proper colouring with the five colours{ }.




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CHROMATICNUMBER

Thechromatic number(G)of a graph is the minimum numberof colours needed to properly colour G.

We have exhibited a 3-colouring so(P)3and as it obviouslycannot be2-coloured (why?), we get(P) =3.




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COLOURING IS HARD

Finding the chromatic number is hard: the decision problem

3-COLOURING

INSTANCE: A graph G

QUESTION: DoesGhave a proper 3-colouring?

isNP-complete

It is also hard inpractice even graphs with a few hundred

vertices are difficult.




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OUTLINE

1 BACKGROUNDBasics

Graph Colouring

4 Colours Suffice


Chromatic Roots

Potts Model

3 COMPLEX ZEROS

Absolute Bounds





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MAP COLOURING

To distinguish regions on a

map in this case a map of

the traditional counties of

England the mapmaker

coloursthem so that two

regions with a common

boundary receive different

colours.




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GUTHRIES OBSERVATION

Around 1852, Francis

Guthrie noticed that he never

needed to use more than 4

colours on any map he triedto colour, and wondered if

that was always the case.

This question became known

as the4-colour conjecture.




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GUTHRIES OBSERVATION

Around 1852, Francis

Guthrie noticed that he never

needed to use more than 4

colours on any map he triedto colour, and wondered if

that was always the case.

This question became known

as the4-colour conjecture.




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FROM MAPS TO GRAPHS

The shapes, sizes and

positions of the regions are

not relevant to the colouring

question and so by replacing

the map with agraphwekeep only the essential

details.

Graphs arising from maps

like this areplanar thereare no crossing edges.




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FROM MAPS TO GRAPHS

The shapes, sizes and

positions of the regions are

not relevant to the colouring

question and so by replacing

the map with agraphwekeep only the essential

details.

Graphs arising from maps

like this areplanar thereare no crossing edges.




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THE FOUR-COLOUR THEOREM

The 4-colour conjecture became the most famous problem in

graph theory, consumed numerous academic careers and had

a far-reaching influence over the development of graph theory.

Finally, more than 120 years later, a heavily computer-aidedproof was published:

THE FOUR COLOUR THEOREM (APPEL & HAKEN 1976)

Every planar graph has a 4-colouring.




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NOT EVERYONE WAS CONVINCED..

Not everyone was convinced

by the proof. . .




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NOT EVERYONE WAS CONVINCED..

Not everyone was convinced

by the proof. . .

. . .and some people remainthat way.


BACKGROUND POLYNOMIALS COMPLEX Z EROS CHROMATIC F UNCTION CHROMATIC ROOTS POTTS M ODEL

O


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OUTLINE

1 BACKGROUNDBasics

Graph Colouring

4 Colours Suffice


Chromatic Roots

Potts Model

3 COMPLEX ZEROSAbsolute Bounds




B


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BIRKHOFF

In 1912 Birkhoff introduced the function PG(q)such that for a

graphGand positive integer q,

PG(q)is thenumberof properq-colourings ofG.

George David Birkhoff

(1884 1944)



Q Q


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QUANTITATIVE VERSUS QUALITATIVE

This was an attempt to developquantitativetools tocountthenumber of colourings of a planar graph, rather than the

alternativequalitativeapproach (Type 1) of just proving the

existence of a 4-colouring.

He hoped to be able to find an analyticproof thatPG(4)> 0forany planar graph G.



EXAMPLE COMPLETE GRAPHS


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EXAMPLE COMPLETEGRAPHS

For the complete graph Kn, where every vertex is joined to all

the others, each vertex must be coloured differently, so the totalnumber ofq-colourings is

PKn(q) =q(q 1)(q 2) . . . (q n+ 1).



EXAMPLE TREES


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EXAMPLE TREES

For atree(i.e. a connected graph with no cycles), there are q

choices of colour for an arbitrary first vertex, and then q 1

choices for each subsequent vertex.

q

q 1

q 1

q 1

q 1

q 1

Thus foranytreeTonnvertices, we have

PT(q) =q(q 1)n1.



ADDITION AND CONTRACTION


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Divide theq-colourings of a graph Gaccording to whether two

(specific) non-adjacent vertices receive the same or differentcolours.

ab

These colourings are in 1-1correspondence with colourings of

G+ab where verticesaand bare

joined by an edge.

abab

These colourings are in 1-1

correspondence with colourings of

G/abwhereaand bare coalesced into

a single vertex (with edges following).





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Divide theq-colourings of a graph Gaccording to whether two

(specific) non-adjacent vertices receive the same or differentcolours.

ab

These colourings are in 1-1correspondence with colourings of

G+ab where verticesaand bare

joined by an edge.

ababab

These colourings are in 1-1

correspondence with colourings of

G/abwhereaand bare coalesced into

a single vertex (with edges following).



DELETION CONTRACTION ALGORITHM


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DELETION-CONTRACTION ALGORITHM

Rearranging this shows that for an edge ab

PG(k) =PGab(k) PG/ab(k)

Repeated application yields thedeletion-contractionalgorithm,

which shows that for ann-vertex graphG:

PG(q)is amonic polynomialof degree n

PG(q)hasalternating coefficients

However this algorithm hasexponential complexityandtherefore is only practical for small graphs.



EXAMPLE PETERSEN GRAPH


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EXAMPLE PETERSEN GRAPH

q1015 q

9 +105 q8455 q7 +1353 q62861 q5 +4275 q44305 q3 +2606 q2704 q

which factors into

q (q 1) (q 2)q

7 12 q

6 + 67 q5 230 q4 + 529 q3 814 q2 + 775 q 352



OUTLINE


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OUTLINE

1 BACKGROUNDBasics

Graph Colouring

4 Colours Suffice


Chromatic Roots

Potts Model





REAL ZEROS


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REAL ZEROS

Birkhoff & Lewis generalized the Five Colour Theorem:

Five-Colour Theorem (Heawood 1890)

IfGisplanarthenPG(5)>0.

Birkhoff-Lewis Theorem (1946)

IfGisplanarandx5, thenPG(x)> 0.

Birkhoff-Lewis Conjecture [still unsolved]

IfGisplanarandx4thenPG(x)> 0.

Leads to studying thereal chromatic zerosof a graph G

maybe studying the real numbers xwherePG(x) = 0will tell uswherePG(x)= 0?



COMPLEX ZEROS


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COMPLEX ZEROS

Why not findallsolutions to the equation

PG(z) = 0?

IfGis an n-vertex graph thenPG(z)has degree nand so thisequation hasnsolutions over the complex numbers these

are thechromatic zerosofG.

First explicit mention of complex chromatic zeros appears to be

in a 1965 paper of Hall, Siry & Vanderslice.



CHROMATIC ZEROS OF 9-VERTEX GRAPHS


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CHROMATIC ZEROS OF 9 VERTEX GRAPHS



FUNDAMENTAL QUESTIONS


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FUNDAMENTAL QUESTIONS

The two fundamental questions prompted by trying to

understand the patterns in plots such as this are:

Are thereabsolute boundson the root-locationindependent of the graph structure?

Can we find bounds on the root-location in terms of graph

parameters?



OUTLINE


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OU

1 BACKGROUND

Basics

Graph Colouring

4 Colours Suffice


Chromatic Roots

Potts Model





POTTS MODEL


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Theq-state Potts modelmodels a physical system as collection

of interacting spins, each taking on one of qdistinct values,

located on a regular lattice grid.

1

2

2

3

4

1

4

3

4

3

2

3

1

2

3

3

4

4

1

4

1

2

4

2 Any possible configuration

:V {1, 2, . . . q}

hasBoltzmann weightgiven by

e=xy

(1 +ve((x), (y)))

[i.e. edgeecontributes 1 +ve if it joinsequal spins]



PARTITION FUNCTION


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Thepartition functionis the sum of the Boltzmann weights:

ZG(q, {ve}) =

:V{1,2,...,q}

e=xy

(1 +ve((x), (y)))

If we put ve=1for every edge physically corresponding to

thezero temperature limit of the antiferromagnetic Potts model then we get

ZG(q,1) =PG(q).

This is no mere accident in one of the amazing examples of theunreasonable effectiveness of mathematics , the full 2-variable partition

function is equivalent to the 2-variable Tutte polynomial of graphs and

matroids.



PHASE TRANSITIONS


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Aphase transitionin a physical system occurs whencontinuous variation in a control parameter yields a

discontinuity in its observed behaviour.

Statistical physicists are interested in complex zeros because a

phase transition can only occur at a real limit pointof thecomplex zerosof the partition function.

Hence azero-free regionfor a family of graphs provides

evidence that phase-transitions cannot occur in that region of

parameter space such theorems are called Lee-Yang

theorems.



POTTS MODELS


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1

2

2

3

4

1

4

3

4

3

2

3

1

2

3

3

4

4

1

4

1

2

4

2

The lattice may beperiodic(i.e.

wrap around).

+

+

+

+

+

+

+

+

+

+

+

+

+

+

Theq= 2case is known as theIsing model.1

1invented by Isings supervisor LenzGORDONR OYLE CHROMATIC P OLYNOMIALS

BACKGROUND POLYNOMIALS COMPLEX Z EROS ABSOLUTEB OUNDS PARAMETERIZED B OUNDS

OUTLINE


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1 BACKGROUND

Basics

Graph Colouring

4 Colours Suffice


Chromatic Roots

Potts Model





ABSOLUTE BOUNDS


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For many years, it was thought that chromatic zeros wererestricted to the right half-plane

Re(z)>0.

Ron Read 2 and I disproved this in 1988 with high-girth cubic

graphsas examples.

For several years after that, papers appeared proving the

existence of chromatic zeros with increasingly large negativereal part.

2For years Ron had a sign on his office door just saying Please ReadGORDONR OYLE CHROMATIC P OLYNOMIALS


CUBIC GRAPHS ON 20 VERTICES


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SOKALS SECOND RESULT


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This game ground to a halt in spectacular fashion, when the

statistical physicist Alan Sokal proved:

THEOREM (SOKAL 2000)

Chromatic zeros are dense in the whole complex plane.

The most surprising part of this theorem is that just one very

simple class of graphs generalized-graphs has

chromatic roots almost everywhere.



ALAN SOKAL


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Sokal is a brilliant statistical physicist and mathematician, but is

most well known for his infamous hoax.

His nonsensical articleTransgressing the Boundaries: Towardsa Transformative Hermeneutics of Quantum Gravityappeared

in the eminent postmodernist journal Social Text and sparked

a furious controversy.



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BACKGROUND

POLYNOMIALS

COMPLEX

ZEROS

ABSOLUTE

BOUNDS

PARAMETERIZED

BOUNDS

OUTLINE


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1 BACKGROUND

Basics

Graph Colouring

4 Colours Suffice


Chromatic Roots

Potts Model




BACKGROUND

POLYNOMIALS

COMPLEX

ZEROS

ABSOLUTE

BOUNDS

PARAMETERIZED

BOUNDS

GENERALIZINGBROOKS THEOREM


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A classic early result of graph theory is Brooks 1941 theoremthat ifGhas maximum degree(G)then

(G)(G) + 1.

Biggs, Damerell & Sands (1971) conjectured the existence of a

functionf(r)such all zeros of PG(z)lie in the region

|z| f(r)

whenever Ghas maximum degree r.



SOKALS FIRST RESULT


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Their conjecture was eventually confirmed:

THEOREM (SOKAL 1999)

IfGis a graph with maximum degree and second largestdegree2 then all zeros ofPG(z)lie in the region

|z| 7.963907

and the region

|z| 7.9639072+ 1

The generalized-graphs show that therecannotbe anybound as a function of3.



THE ANSWER


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I have been convinced for over 20 years that the realanswer is

CONJECTURE (ROYLE C.1990)

The complete bipartite graphKr,rcontributes the chromatic root

of maximum modulus among all graphs of maximum degree r

(excluding K4 forr=3) a bound of around 1.6.

Thus the chromatic root of K4,4 at

z=2.802489 + 3.097444i

should have largest modulus overall graphs

with = 4.



10 VERTEX GRAPHS WITH = 3


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10 VERTEX GRAPHS WITH = 4


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QUARTIC GRAPHS ON 15 AND 16 VERTICES


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MANY QUESTIONS REMAIN


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I am currently working with Sokal and others to try to improve

the bounds onrealchromatic roots.

The following sequence of (increasingly strong) conjecturesare

all open: ifGhas maximum degreethen

1 PG(q)>0for allq>;

2 PG(q)and its derivativesare positive for q>;

3 All roots ofPG(q),real or complex, have real part at most.

Successfully attacking these problems requirescomputation,graph theory, some basiccomplex analysis, a talent forpattern

spottingand a bit ofluck!



CUBIC GRAPHS


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Whichcubic graphhas the largest real chromatic root?

Computation shows that among the cubic graphs on up to 20vertices (about 1/2 million of them), this one graph is the

record holderwith a real chromatic root at (about) 2.77128607.

Canyoubreak the record?


chromatic polynomials

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