1912-2015: 100+ years to chromatic polynomial outline: origins of the connection-contraction the...
TRANSCRIPT
![Page 1: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/1.jpg)
1912-2015: 100+ years to Chromatic
Polynomial
![Page 2: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/2.jpg)
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
![Page 3: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/3.jpg)
Mathematical commercial:Mathematical commercial:
04/18/2304/18/23 33People + Ideas = HistoryPeople + Ideas = History
![Page 4: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/4.jpg)
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.
04/18/23 People + Ideas = History 4
![Page 5: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/5.jpg)
George David Birkhoff: 1884-1944
04/18/23 People + Ideas = History 5
• 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
![Page 6: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/6.jpg)
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.
04/18/23 People + Ideas = History 6
![Page 7: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/7.jpg)
George David Birkhoff: 1884-1944
04/18/23 People + Ideas = History 7
• 1935: Menzel, Einstein and Birkhoff,
• the time of Einstein's receiving an honorary degree from Harvard; Cambridge, Massachusetts;
•Date: Unknown
![Page 8: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/8.jpg)
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.
04/18/23 People + Ideas = History 8
![Page 9: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/9.jpg)
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
04/18/23 People + Ideas = History 9
![Page 10: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/10.jpg)
04/18/23 People + Ideas = History 10
1912:
![Page 11: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/11.jpg)
04/18/23 People + Ideas = History 11
1912:
![Page 12: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/12.jpg)
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?
04/18/23 People + Ideas = History 12
![Page 13: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/13.jpg)
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).
04/18/23 People + Ideas = History 13
![Page 14: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/14.jpg)
Hassler Whitney: 1907-1989
04/18/23 People + Ideas = History 14
1932: published the paper “The colorings of graphs” (Ann. Of Math. 33 (1932) 688-718
![Page 15: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/15.jpg)
Hassler Whitney: 1907-1989
04/18/23 People + Ideas = History 15
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.
![Page 16: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/16.jpg)
04/18/23 People + Ideas = History 16
1932:
![Page 17: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/17.jpg)
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”
04/18/23 People + Ideas = History 17
![Page 18: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/18.jpg)
04/18/23 People + Ideas = History 18
1932:
![Page 19: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/19.jpg)
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
04/18/23 People + Ideas = History 19
![Page 20: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/20.jpg)
04/18/23 People + Ideas = History 20
1949:
![Page 21: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/21.jpg)
Alexander Zykov (1922-2013)
04/18/23 People + Ideas = History 21
Odessa, Ukraine, 2009
![Page 22: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/22.jpg)
With Zykov and Vizing 9/11/2001
04/18/23 People + Ideas = History 22
![Page 23: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/23.jpg)
Connection-contraction algorithm:
04/18/23 People + Ideas = History 23
= +
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
![Page 24: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/24.jpg)
Deletion-contraction algorithm:
04/18/23 People + Ideas = History 24
= -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
![Page 25: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/25.jpg)
Zykov ß-polynomial (1976):
04/18/23 People + Ideas = History 25
= + ß
G G1 G2
contractionconnectionnon adjacent
H(G, ß) =1 +a1ß+a2ß2 +… = combinations of complete graphs…
a b a b ab
![Page 26: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/26.jpg)
Zykov ß-polynomial (1976)
04/18/23 People + Ideas = History 26
x n
nK
nH
1nH
nn HDH ]1[ 21
Differentiation operator for ß-polynomial
(never published)
Add simplicial vertex of degree n-ν
![Page 27: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/27.jpg)
Coefficients of P(G,λ)
04/18/23 People + Ideas = History 27
![Page 28: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/28.jpg)
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
04/18/23 People + Ideas = History 28
n
i
ii GrGP
1
)()(),(
)(Grii
)1)...(2)(1()( ii
![Page 29: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/29.jpg)
Coefficients of P(G,λ):• Whitney, 1932, the broken circuit theorem: • if then
04/18/23 People + Ideas = History 29
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).
![Page 30: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/30.jpg)
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
04/18/23 People + Ideas = History 30
![Page 31: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/31.jpg)
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)
04/18/23 People + Ideas = History 31
k
||||...||...|||| 121 nnk aaaaa
![Page 32: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/32.jpg)
The roots of P(G,λ) for any G
04/18/23 People + Ideas = History 32
![Page 33: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/33.jpg)
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.
04/18/23 People + Ideas = History 33
![Page 34: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/34.jpg)
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
![Page 35: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/35.jpg)
The roots of P(G,λ) for any G
• 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…
04/18/23 People + Ideas = History 35
![Page 36: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/36.jpg)
The roots of P(G,λ) for any G
• 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.
04/18/23 People + Ideas = History 36
![Page 37: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/37.jpg)
The roots of P(G,λ) for planar G
04/18/23 People + Ideas = History 37
![Page 38: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/38.jpg)
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
04/18/23 People + Ideas = History 38
![Page 39: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/39.jpg)
The roots of P(G,λ) for planar G
• 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!
04/18/23 People + Ideas = History 39
![Page 40: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/40.jpg)
The roots of P(G,λ) for planar G
• 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)
04/18/23 People + Ideas = History 40
540 1NO roots
?
NO roots32/27 2 3 ?
? ? ?
![Page 41: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/41.jpg)
The all-integer roots of P(G,λ) (for any G)
04/18/23 People + Ideas = History 41
![Page 42: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/42.jpg)
The all-integer roots of P(G,λ) (for any G)
04/18/23 People + Ideas = History 42
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”…
![Page 43: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/43.jpg)
The all-integer roots of P(G,λ) (for any G)
• 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
![Page 44: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/44.jpg)
The integer roots of P(G,λ) (for any G)
04/18/23 People + Ideas = History 44
mK
xG
G
),()(),( xGPmGP
x
Simplicial vertex of degree m
The root is the degree of a simplicial vertex when we delete it
m
![Page 45: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/45.jpg)
The all-integer roots of P(G,λ)
04/18/23 People + Ideas = History 45
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
![Page 46: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/46.jpg)
Tutte Polynomial
04/18/23 People + Ideas = History 46
![Page 47: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/47.jpg)
William Tutte (1917-2002)
04/18/23 People + Ideas = History 47
19991936
![Page 48: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/48.jpg)
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.
04/18/23 People + Ideas = History 48
![Page 49: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/49.jpg)
Tutte polynomial:
),,( yxGT
04/18/23 People + Ideas = History 49
)( eGxT
)( eGyT
)()( eGTeGT
if is a bridgee
if is a loope
otherwise
1 if nEG
![Page 50: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/50.jpg)
Tutte polynomial:• For a general undirected graph
04/18/23 People + Ideas = History 50
:),( EVG
||||)()()( )1()1(),,( VAAc
EA
EcAc yxyxGT
Where is the number of connected components of graph
)(Ac
),( AV
![Page 51: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/51.jpg)
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)
04/18/23 People + Ideas = History 51
![Page 52: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/52.jpg)
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
04/18/23 People + Ideas = History 52
![Page 53: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/53.jpg)
Tutte Polynomial contains the Chromatic Polynomial:
),( GP
04/18/23 People + Ideas = History 53
)0,1,()1( )()(|| GTGcGcV
![Page 54: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/54.jpg)
Potts model
04/18/23 People + Ideas = History 54
![Page 55: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/55.jpg)
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.
04/18/23 People + Ideas = History 55
He 01
kT
T
![Page 56: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/56.jpg)
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
04/18/23 People + Ideas = History 56
,q
0T
![Page 57: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/57.jpg)
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.
04/18/23 People + Ideas = History 57
![Page 58: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/58.jpg)
Tutte Plane for
04/18/23 People + Ideas = History 58
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
![Page 59: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/59.jpg)
Chromatic polynomial of Mixed Hypergraphs
04/18/23 People + Ideas = History 59
![Page 60: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/60.jpg)
The next turn: mixed hypergraph coloring (1993):
04/18/23 People + Ideas = History 60
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
![Page 61: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/61.jpg)
Connection-contraction becomes Splitting-Contraction:
04/18/23 People + Ideas = History 61
=
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
![Page 62: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/62.jpg)
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:
04/18/23 People + Ideas = History 62
)(' Hr
'
)()(),(
i
ii HrHP
It was n since Birkhoff
![Page 63: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/63.jpg)
New properties of P(H,λ)• Mixed hypergraph H may have PHANTOM
(invisible) VERTICES (never happened!):
• The chromatic spectrum may be broken (never happened):
•
04/18/23 People + Ideas = History 63
),(),( xHPHP
)0,...,0,,...0,...,,0,...,0()( ' rrHR
First positive component Last positive component
GAP
![Page 64: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/64.jpg)
• 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
04/18/23 People + Ideas = History 64
Philosophy behind the concept
![Page 65: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/65.jpg)
WHO KNEW!
04/18/23 People + Ideas = History 65
![Page 66: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/66.jpg)
The newest application: Byzantine agreement and cyber-security
04/18/23 People + Ideas = History 66
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.
![Page 67: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/67.jpg)
The newest application: Byzantine agreement and cyber-security
04/18/23 People + Ideas = History 67
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.
![Page 68: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/68.jpg)
The newest application: Byzantine agreement and cyber-security
• 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
![Page 69: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/69.jpg)
The newest application: Byzantine agreement and cyber-security
04/18/23 People + Ideas = History 69
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”.
![Page 70: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/70.jpg)
The newest application: Byzantine agreement and cyber-security
04/18/23 People + Ideas = History 70
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:
![Page 71: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/71.jpg)
The newest application: Byzantine agreement and cyber-security
04/18/23 People + Ideas = History 71
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.
![Page 72: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/72.jpg)
The newest application: Byzantine agreement and cyber-security
04/18/23 People + Ideas = History 72
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).
![Page 73: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/73.jpg)
The paper itself:
04/18/23 People + Ideas = History 73
![Page 74: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/74.jpg)
PhD Thesis at Princeton
04/18/23 People + Ideas = History 74
![Page 75: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/75.jpg)
PhD Thesis at Princeton
04/18/23 People + Ideas = History 75
![Page 76: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/76.jpg)
WHO KNEW??
•PAUL ERDÖS KNEW IT!!!
04/18/23 People + Ideas = History 76
![Page 77: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/77.jpg)
04/18/23 People + Ideas = History 77
![Page 78: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/78.jpg)
Marceille 1995
04/18/23 People + Ideas = History 78
![Page 79: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/79.jpg)
WHO KNEW?!
04/18/23 People + Ideas = History 79
![Page 80: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/80.jpg)
The very first question was …
04/18/23 People + Ideas = History 80
Herbert S. Wilf: 1931-2012
![Page 81: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/81.jpg)
One of the papers was …
04/18/23 People + Ideas = History 81
![Page 82: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/82.jpg)
CONCLUSION
04/18/23 People + Ideas = History 82
![Page 83: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/83.jpg)
Mixed Hypergraph Coloring today
04/18/23 People + Ideas = History 83
![Page 84: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/84.jpg)
Possible problems for research:
• Differentiation of chromatic plynomials• The eigenvalues of self dual hypergraphs• Mixed Ramsey Hypergraphs - spectrum
04/18/23 People + Ideas = History 84
![Page 85: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/85.jpg)
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.
04/18/23 People + Ideas = History 85
![Page 86: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/86.jpg)
What a world!!!
04/18/23 People + Ideas = History 86
![Page 87: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/87.jpg)
04/18/23 People + Ideas = History 87
THANK YOU!For contribution to the theory of chromatic polynomial by inviting me to give this talk and actually attending it!
![Page 88: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/88.jpg)
Memories…
04/18/23 People + Ideas = History 88
![Page 89: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/89.jpg)
Final thanks to …
04/18/23 People + Ideas = History 89
![Page 90: 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](https://reader037.vdocuments.mx/reader037/viewer/2022110321/56649d035503460f949d6752/html5/thumbnails/90.jpg)
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.
04/18/23 People + Ideas = History 90