Post on 04-Jun-2018
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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
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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
Parameterized Bounds
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BACKGROUND POLYNOMIALS COMPLEX Z EROS BASICS GRAPH C OLOURING 4 COLOURS S UFFICE
GRAPHS
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BACKGROUND POLYNOMIALS COMPLEX Z EROS BASICS GRAPH C OLOURING 4 COLOURS S UFFICE
GRAPHS
Vertices
GORDONR OYLE CHROMATIC P OLYNOMIALS
C G C 4 C S
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BACKGROUND POLYNOMIALS COMPLEX Z EROS BASICS GRAPH C OLOURING 4 COLOURS S UFFICE
GRAPHS
Vertices
Edges
GORDONR OYLE CHROMATIC P OLYNOMIALS
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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BACKGROUND POLYNOMIALS COMPLEX Z EROS BASICS GRAPH C OLOURING 4 COLOURS S UFFICE
BASIC TERMINOLOGY
Petersen Graph
Phas 10vertices
Phas 15edges
Each vertex has 3 neighbours
Pis3-regular(akacubic)
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BACKGROUND POLYNOMIALS COMPLEX Z EROS BASICS GRAPH C OLOURING 4 COLOURS S UFFICE
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BACKGROUND POLYNOMIALS COMPLEX Z EROS BASICS GRAPH C OLOURING 4 COLOURS S UFFICE
POSITION IS IRRELEVANT
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS BASICS GRAPH C OLOURING 4 COLOURS S UFFICE
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BACKGROUND POLYNOMIALS COMPLEX Z EROS BASICS GRAPH C OLOURING 4 COLOURS S UFFICE
POSITION IS IRRELEVANT
The Petersen graph isnot planar
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS BASICS GRAPH C OLOURING 4 COLOURS S UFFICE
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BACKGROUND POLYNOMIALS COMPLEX Z EROS BASICS GRAPH C OLOURING 4 COLOURS S UFFICE
POSITION IS IRRELEVANT
The Petersen graph isnot planarandnot hamiltonian
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS BASICS GRAPH C OLOURING 4 COLOURS S UFFICE
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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 POLYNOMIALSChromatic Function
Chromatic Roots
Potts Model
3 COMPLEX ZEROS
Absolute Bounds
Parameterized Bounds
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS BASICS GRAPH C OLOURING 4 COLOURS S UFFICE
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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{ }.
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS BASICS GRAPH C OLOURING 4 COLOURS S UFFICE
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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.
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS BASICS GRAPH C OLOURING 4 COLOURS S UFFICE
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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.
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS BASICS GRAPH C OLOURING 4 COLOURS S UFFICE
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OUTLINE
1 BACKGROUNDBasics
Graph Colouring
4 Colours Suffice
2 POLYNOMIALSChromatic Function
Chromatic Roots
Potts Model
3 COMPLEX ZEROS
Absolute Bounds
Parameterized Bounds
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS BASICS GRAPH C OLOURING 4 COLOURS S UFFICE
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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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BACKGROUND POLYNOMIALS COMPLEX Z EROS BASICS GRAPH C OLOURING 4 COLOURS S UFFICE
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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.
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS BASICS GRAPH C OLOURING 4 COLOURS S UFFICE
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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.
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS BASICS GRAPH C OLOURING 4 COLOURS S UFFICE
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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.
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS BASICS GRAPH C OLOURING 4 COLOURS S UFFICE
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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.
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS BASICS GRAPH C OLOURING 4 COLOURS S UFFICE
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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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BACKGROUND POLYNOMIALS COMPLEX Z EROS BASICS GRAPH C OLOURING 4 COLOURS S UFFICE
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NOT EVERYONE WAS CONVINCED..
Not everyone was convinced
by the proof. . .
. . .and some people remainthat way.
GORDONR OYLE CHROMATIC P OLYNOMIALS
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
2 POLYNOMIALSChromatic Function
Chromatic Roots
Potts Model
3 COMPLEX ZEROSAbsolute Bounds
Parameterized Bounds
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS CHROMATIC F UNCTION CHROMATIC ROOTS POTTS M ODEL
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)
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS CHROMATIC F UNCTION CHROMATIC ROOTS POTTS M ODEL
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.
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS CHROMATIC F UNCTION CHROMATIC ROOTS POTTS M ODEL
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).
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS CHROMATIC F UNCTION CHROMATIC ROOTS POTTS M ODEL
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.
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS CHROMATIC F UNCTION CHROMATIC ROOTS POTTS M ODEL
ADDITION AND CONTRACTION
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ADDITION AND CONTRACTION
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).
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS CHROMATIC F UNCTION CHROMATIC ROOTS POTTS M ODEL
ADDITION AND CONTRACTION
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ADDITION AND CONTRACTION
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).
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS CHROMATIC F UNCTION CHROMATIC ROOTS POTTS M ODEL
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.
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS CHROMATIC F UNCTION CHROMATIC ROOTS POTTS M ODEL
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
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS CHROMATIC F UNCTION CHROMATIC ROOTS POTTS M ODEL
OUTLINE
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OUTLINE
1 BACKGROUNDBasics
Graph Colouring
4 Colours Suffice
2 POLYNOMIALSChromatic Function
Chromatic Roots
Potts Model
3 COMPLEX ZEROSAbsolute Bounds
Parameterized Bounds
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS CHROMATIC F UNCTION CHROMATIC ROOTS POTTS M ODEL
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?
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS CHROMATIC F UNCTION CHROMATIC ROOTS POTTS M ODEL
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.
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS CHROMATIC F UNCTION CHROMATIC ROOTS POTTS M ODEL
CHROMATIC ZEROS OF 9-VERTEX GRAPHS
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CHROMATIC ZEROS OF 9 VERTEX GRAPHS
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS CHROMATIC F UNCTION CHROMATIC ROOTS POTTS M ODEL
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?
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS CHROMATIC F UNCTION CHROMATIC ROOTS POTTS M ODEL
OUTLINE
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OU
1 BACKGROUND
Basics
Graph Colouring
4 Colours Suffice
2 POLYNOMIALSChromatic Function
Chromatic Roots
Potts Model
3 COMPLEX ZEROSAbsolute Bounds
Parameterized Bounds
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS CHROMATIC F UNCTION CHROMATIC ROOTS POTTS M ODEL
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]
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS CHROMATIC F UNCTION CHROMATIC ROOTS POTTS M ODEL
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.
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS CHROMATIC F UNCTION CHROMATIC ROOTS POTTS M ODEL
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.
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS CHROMATIC F UNCTION CHROMATIC ROOTS POTTS M ODEL
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
2 POLYNOMIALSChromatic Function
Chromatic Roots
Potts Model
3 COMPLEX ZEROSAbsolute Bounds
Parameterized Bounds
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS ABSOLUTEB OUNDS PARAMETERIZED B OUNDS
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
BACKGROUND POLYNOMIALS COMPLEX Z EROS ABSOLUTEB OUNDS PARAMETERIZED B OUNDS
CUBIC GRAPHS ON 20 VERTICES
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GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS ABSOLUTEB OUNDS PARAMETERIZED B OUNDS
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.
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS ABSOLUTEB OUNDS PARAMETERIZED B OUNDS
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.
GORDONR OYLE CHROMATIC P OLYNOMIALS
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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
2 POLYNOMIALSChromatic Function
Chromatic Roots
Potts Model
3 COMPLEX ZEROSAbsolute Bounds
Parameterized Bounds
GORDONR OYLE CHROMATIC P OLYNOMIALS
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.
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS ABSOLUTEB OUNDS PARAMETERIZED B OUNDS
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.
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS ABSOLUTEB OUNDS PARAMETERIZED B OUNDS
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.
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS ABSOLUTEB OUNDS PARAMETERIZED B OUNDS
10 VERTEX GRAPHS WITH = 3
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GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS ABSOLUTEB OUNDS PARAMETERIZED B OUNDS
10 VERTEX GRAPHS WITH = 4
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GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS ABSOLUTEB OUNDS PARAMETERIZED B OUNDS
QUARTIC GRAPHS ON 15 AND 16 VERTICES
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GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS ABSOLUTEB OUNDS PARAMETERIZED B OUNDS
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!
GORDONR OYLE CHROMATIC P OLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX Z EROS ABSOLUTEB OUNDS PARAMETERIZED B OUNDS
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?
GORDONR OYLE CHROMATIC P OLYNOMIALS