Transcript Chromatic Roots and Fibonacci Numbers By: Saeid Alikhani

Chromatic Roots and Fibonacci Numbers
Saeid Alikhani and Yee- hock Peng
Institute for Mathematical Research
University Putra Malaysia
Workshop
“Zeros of Graph Polynomials”
Isaac Newton Institute for Mathematical Science, Cambridge University, UK
21-25 January 2008
• Outline of Talks
• 1. Introduction
• 2. Chromatic roots and golden ratio
• 3. Chromatic roots and n-anacci constant
• 4. Some questions
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• Introduction
• A graph G consists of
set V (G) of vertices, and
set E(G) of unordered pairs of vertices called edges.
• These graphs are undirected
• A graph is planar if it can be drawn in the plane with no edges
crossing.
• A (proper) k-colouring of a graph G is a mapping
, , k} where f (u )  f (v)
for every edge
f : V (G)  {1,2,
uv  E (G)
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• The four colour Theorem:
• Probably the most famous result in graph theory is the following
• theorem:
• Four-Colour Theorem:
• Every planar graph is 4-colourable.
• Near-triangulation graphs: plane graphs with at most one nontriangular face.
• A near- triangulation with 3-face is a triangulation.
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• The number of distinct k-colourings of G, denoted by P(G;k) is
called the chromatic polynomial of G.
•
A root of P(G;k) is called a chromatic root of G.
• An interval is called a root-free interval for a chromatic polynomial
P(G; k) if G has no chromatic root in this interval.
• (Birkhoff and Lewis 1946): (-∞,0), (0,1) , (1,2) and [5,∞) are zero
free intervals for all plane triangulations graph.
• Chromatic Zero-free intervals: (-∞,0), (0,1)
• (Jackson 1993): (1,32/27] is also a chromatic zero-free interval.
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• (Thomassen 1997) There are no more chromatic zero-free intervals.
• We recall that a complex number is called an algebraic number
(resp. an algebraic integer) if it is a root of some monic polynomial
with rational (resp. integer) coefficients.
• Corresponding to any algebraic number  , there is a unique monic
polynomial p with rational coefficients, called the minimal polynomial
of  (over the rationals), with the property that p divides every
polynomial with rational coefficients having  as a root.
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• Two algebraic numbers  and  are called conjugate if they have
the same minimal polynomial.
*
• Since the chromatic polynomial P(G; k) is a monic polynomial in k
with integer coefficients, its roots are, by definition, algebraic
integers. This naturally raises the question:
• Which algebraic integers can occur as roots of chromatic
polynomials?
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32 

• Clearly those lying in ( ,0)  (0,1)  1,
are forbidden set.

 27 
•
Using this reasoning, Tutte [13] proved that
3 5
the Beraha number B5 
2
cannot be a chromatic root.
Salas and Sokal in [10] extended this result to show that the
generalized beraha numbers
Bn
(k )
k 

 4 cos  
 n 
2
, for
n  5,7,8,9
and n  11 , with k coprime to n, are never chromatic roots. For n =
5 5
5 5
*
10 they showed the weaker result that B10 
and B 10 
2
2
are not chromatic roots of any plane near-triangulation.
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• Fibonacci numbers are terms of the sequence defined in a quite
simple recursive fashion.
• However, despite its simplicity, they have some curious properties
which are worth attention.
•
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1,1,2,3,5,8,13,21,…
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•
Golden Ratio and Chromatic Roots
F1  F2  1 and Fn  Fn 1  Fn  2
Fn 1
Golden ratio:   lim
n F
n
1 n
Theorem1: For every natural number n, Fn 
(  (1   ) n ).
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Corollary 1: If n is even Fn   and if n is odd Fn   .
Fn1
Fn 1
2
Theorem 2: [Cassini`s Formula]: Fn 1 Fn 1  Fn  (1) n .
• Fibonacci sequence:
•
•
•
•
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• Theorem 3: For every natural n,
 1
| Fn1   Fn |  0, 
 Fn 
1
Proof. Suppose that n is even, therefore n-1 is odd, and by
Fn 1
Fn  2
1
Fn 1
Fn
 
Corollary 1, we have
,and hence
 
Fn
Fn 1
Fn  2
Fn 1
and by multiplying Fn in this inequality, we have
Fn  2 Fn
Fn 1   Fn 
Fn 1
1
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1
• Thus by Theorem 2, we have 0  Fn 1   Fn 
. Similarly,
Fn 1
the result holds when n is odd.
n

 Fn  Fn1 (n  2) .
• Theorem 4.( [7], P.78):
1
The following theorem is a consequence of Salas-Sokal Proposition
in [10]:
• Theorem 5. Consider a number of the form  
q are rational, r  2 is an integer that is not a
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perfect square, and p  | q | r  .
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any chromatic polynomial.
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Then
p  q r , where p,

is not the root of
12
• Proof. If   p  q r is a root of some polynomial with integer
coeffcients (e.g. a chromatic polynomial), then so its conjugate
*  p  q r .But  or * can lie in (,0)  (0,1)  (1, 32 ]
a contradiction.
27
n
• Corollary 2: For every natural n,  cannot be a root of any
chromatic polynomials.
n
Proof. By Theorem 4, we can consider  of the form   p  q
with
Fn
p   Fn 1
2
and
Fn
q
2
r
1
p

|
q
|
r



Fn  Fn 1
. Since
1
32

by Theorem 3, p  | q | r 
.Therefore we have the
Fn 1 27
result by Theorem 5.
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•
•
Chromatic Roots and n-anacci Constant
An n-step (
for k>2
F1
n  2 ) Fibonacci sequence :
(n)
 F2
(n)
 1, Fk
(n)
n
  Fk i
(n)
i 1
(n)
Fk
n  lim
(n)
k 
Fk 1
• It is easy to see that  n is the real positive root of
• n-anacci constant:
f n ( x)  x n  x n 1    x  1
• Also note that lim  n  2 . (See [8], [14]).
n 
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• Theorem 6. ([8]) The polynomial f n ( x) 
an irreducible polynomial over Q. 
x n  x n 1    x  1 is
• Theorem 7. ([4]) Let G be a graph with n vertices and k connected
components. Then the chromatic polynomial of G is of the form
P(G,  )  an n  an 1n 1    ak k
with an , an 1 ,  , ak integer, a  1 , and (1)
n
Furthermore, if G has at least one edge, then
P(G,1)  a  a    a  0 .
n
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n 1
n l
al  0, k  l  n
k
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• Theorem 8. For every natural n, the 2n-anacci numbers cannot be
roots of any chromatic polynomials.
Proof. We know that  2 n is a root of f 2 n ( x)  x 2 n  x 2 n 1    x  1
which is minimal polynomial for this root. It is obvious that f 2 n ( x ) is
not a chromatic polynomial. Now suppose that there exist a
chromatic polynomial P(x) such that P ( 2 n )  0
. By Theorem
6, f 2 n ( x) | P ( x) . Since f 2 n (0)  1  0 , and f 2 n (1)  1  0
by the intermediate value theorem, f 2 n ( x) and therefore P(x) has a
root in (-1, 0) and this is a contradiction.
• Theorem 9. All natural powers of  2 n cannot be chromatic root.
m
• Proof. Suppose that 2 n (m  N ) is a chromatic root, that is there
exist a chromatic polynomial P(G,  )  a n  a n 1    a 
n
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n 1
1
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Such that
P(G,2 n )  0. Therefore
m
2 n  ak 12 n
   a12 n  0
So we can say that  2 n is a root of the polynomial,
Q( )  mk  ak 1mk m    a1
mk
m ( k 1)
m
f 2 n ( ) | Q ( ) . Since f 2 n (0)  1  0 and f 2 n (1)  1  0m
f 2 n ( ) and so Q( )have a root say  in (-1,0). Therefore 
is a root of P(G,  ) . Since  m  ( 1,0)  (0,1) , we have a
so
contradiction.
How about (2n+1)- anacci??
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• We think that (2n + 1)-anacci numbers and all natural power of them
also cannot be chromatic roots, but we are not able to prove it!
n
• Theorem 10. (Dong et al [4]) Let P(G,  )   ai  be a chromatic
i 1
1  i  n 1
polynomial of a graph G of order n. Then for any
i
n 

| ai 1 |   1ai (where equality holds if and only if G is a tree).
i

• Theorem 11. For every natural n,  2 n 1 can not be a root of
chromatic polynomial of graph G with at most 4n + 2 vertices.
• Proof. We know that (2n+1)- anacci is a root of
f 2 n1 ( x)  x 2 n1  x 2 n    x  1
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• Now suppose that there exist a chromatic polynomial
P( x)  am x m   a1 x  a0 , (m  2n  1) , such that, P( 2 n 1 )  0
Therefore, there exist g ( x)  b
x m2 n 1    b x  b , such
m  2 n 1
that
1
0
P( x)  f 2 n1 ( x) g ( x) . We have for m  2n  1  2n  1
a0  b0  0, a1  b1
a2  b2  b1 ,

am  2 n 1  bm  2 n 1    b1
• By Theorem 7,
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( x  1) | P( x), and so
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m  2 n 1
x

1
|
b
x
   b1 x  b0 . By the above equalities, we
•
m  2 n 1
have am  2 n 1  0 . By Theorem 10,
m
| am | 
 1 | am 1 |  
 m 1 
m
 m  1
|a

1

 
 m  2 n 1 | 0
 m  1   m  2n  1 
• So, we have am  2 n    am 1  am  0 . Therefore P(x) cannot
.
be a chromatic polynomial, and this is a contradiction 
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• Some questions and remarks
• (Jackson 1993) For any ε>0, there exists a graph G such that
P(G, λ) has a zero in (32/27, 32/27+ ε).
• Theorem above says what is on the right of the number of 32/27 in
general case. But the problem has been considered for some
families of graphs as well. One of this families is triangulation
graphs, and there are some open problems for it. We recall the
Beraha question, which says:
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• Question 1. (Beraha's question [1]) Is it true for every   0 , there
exists a plane triangulation G such that P(G,  ) has a root in,
2
( Bn   , Bn   ) , where Bn  2  2 cos is called the n-th
n
Beraha constant (or number)?
• Beraha et al. [3] proved that B   2  1    2.61803
5
is an accumulation point of real chromatic roots of certain plane
triangulations.
• Jacobsen et al. [6] extended this to show that B7 , B8 , B9 and B10
are likewise accumulation points of real chromatic roots of plane
triangulations.
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• Finally, Royle [9] has recently exhibited a family of plane
triangulations with chromatic roots converging to 4.
• Of course, it is an open question which other numbers in the interval
( 32/27, 4) can be accumulation points of real chromatic roots of
planar graphs.
• The following conjecture of Thomassen is one possible answer.
• Conjecture 1. The set of chromatic roots of the family of planar
graphs consists of 0,1 and a dense subset of ( 32/27,4).(See [12]).
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• Now, let An .i  i   n , where i {0,1,2} .
• Here we ask the following question that is analogous to Beraha's
question.
• Question 2. Is it true that, for any  0 , there exists a plane
triangulation graph G such that P(G,  ) has a root in
( An ,i   , An ,i   ) ?( n  2, i {0,1,2}).
• Note that A2 , 2  B10  2   . Beraha, Kahane and Reid [2] proved
that the answer to our question (or respectively, Beraha question) is
positive for i,n=2 (or n = 10).
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• References
• [1] Beraha, Infinite non-interval families of maps and chromials,
Ph.D. thesis, Johns Hopkins University, 1975.
• [2] Beraha,S., Kahane, J, and R. Reid, B7 and B10 are limit points of
chromatic zeros, Notices Amer. Math. Soc. 20(1973), 45.
• [3] Beraha,S., Kahane,J. and Weiss, N.J., Limits of chromatic zeros
of some families of maps, J. Combinatorial Theory Ser. B 28 (1980),
52-65.
• [4] Dong, F.M, Koh, K. M, Teo, K. L, Chromatic polynomial and
chromaticity of graphs, World Scienti¯c Publishing Co. Pte. Ltd.
2005.
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• [5] Jackson, B., A zero free interval for chromatic polynomials of
graphs, Combin.Probab. Comput. 2 (1993) 325-336.
• [6] J.L. Jacobsen, J. Salas and A.D. Sokal, Transfer matrices and
partition-function zeros for antiferromagnetic Potts models. III.
Triangular-lattice chromatic polynomial,J.Statist. Phys. 112 (2003),
921-1017, see e.g. Tables 3 and 4.
• [7] Koshy, T., Fibonacci and Lucas numbers with applications, A
Willey-Interscience Publication, 2001.
• [8] Martin, P. A, The Galois group of x 2 n1  x 2 n    x  1 ,
Journal of pure and applied algebra. 190 (2004) 213-223.
• [9] G. Royle, Planar triangulations with real chromatic roots
arbitrarily close to four, http://arxiv.org/abs/math.CO/0511304.
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• [10] J. Salas and A.D. Sokal, Transfer matrices and partition-function
zeros for antiferromagnetic Potts models. I. General theory and
square-lattice chromatic polynomial,J. Statist. Phys. 104 (2001),
609-699.
• [11] I. Stewart and D. Tall, Algebraic Number Theory, 2nd ed,
Chapman and Hall,London- New York, 1987.
• [12] Thomassen, C, The zero- free intervals for chromatic
polynomials of graphs, Combin.Probab. Comput. 6,(1997), 497-506.
• [13] Tutte, W.T., On chromatic polynomials and golden ratio, J.
Combinatorial Theory,Ser B 9 (1970),289-296.
• [14] http://mathworld.wolfram.com/Finonaccin-StepNumber.html (last
accessed on Dec2006)
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•
Thanks for your attention!
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