Chromatic Roots and Fibonacci Numbers Saeid Alikhani and
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 Saeid Alikhani, UPM
• Outline of Talks • 1. Introduction • 2. Chromatic roots and golden ratio • 3. Chromatic roots and n-anacci constant • 4. Some questions Saeid Alikhani, UPM 2
• 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 , where for every edge Saeid Alikhani, UPM 3
• 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. Saeid Alikhani, UPM 4
• 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. Saeid Alikhani, UPM 5
• (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. Saeid Alikhani, UPM 6
• Two algebraic numbers and the same minimal polynomial. are called conjugate if they have • 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? Saeid Alikhani, UPM 7
• Clearly those lying in • are forbidden set. Using this reasoning, Tutte [13] proved that the Beraha number cannot be a chromatic root. Salas and Sokal in [10] extended this result to show that the generalized beraha numbers and , for , with k coprime to n, are never chromatic roots. For n = 10 they showed the weaker result that and are not chromatic roots of any plane near-triangulation. Saeid Alikhani, UPM 8
• 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. • Saeid Alikhani, UPM 1, 1, 2, 3, 5, 8, 13, 21, … 9
• Golden Ratio and Chromatic Roots • Fibonacci sequence: and • Golden ratio: • Theorem 1: For every natural number n, • Corollary 1: If n is even • Theorem 2: [Cassini`s Formula]: Saeid Alikhani, UPM and . � if n is odd . �. � 10
• Theorem 3: For every natural n, Proof. Suppose that n is even, therefore n-1 is odd, and by Corollary 1, we have and by multiplying Saeid Alikhani, UPM , and hence in this inequality, we have 11
• Thus by Theorem 2, we have. Similarly, the result holds when n is odd. � • Theorem 4. ( [7], P. 78): . � The following theorem is a consequence of Salas-Sokal Proposition in [10]: • Theorem 5. Consider a number of the form q are rational, is an integer that is not a perfect square, and any chromatic polynomial. Saeid Alikhani, UPM . Then , where p, is not the root of 12
• Proof. If is a root of some polynomial with integer coeffcients (e. g. a chromatic polynomial), then so its conjugate. But or can lie in a contradiction. � • Corollary 2: For every natural n, cannot be a root of any chromatic polynomials. Proof. By Theorem 4, we can consider of the form with by Theorem 3, result by Theorem 5. � Saeid Alikhani, UPM and . Since . Therefore we have the 13
• • Chromatic Roots and n-anacci Constant An n-step ( ) Fibonacci sequence : for k>2 • n-anacci constant: • It is easy to see that • Also note that Saeid Alikhani, UPM is the real positive root of . (See [8], [14]). 14
• Theorem 6. ([8]) The polynomial an irreducible polynomial over Q. � 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 with integer, , and Furthermore, if G has at least one edge, then. � Saeid Alikhani, UPM 15
• Theorem 8. For every natural n, the 2 n-anacci numbers cannot be roots of any chromatic polynomials. Proof. We know that is a root of which is minimal polynomial for this root. It is obvious that is not a chromatic polynomial. Now suppose that there exist a chromatic polynomial P(x) such that. By Theorem 6, . Since , and by the intermediate value theorem, and therefore has a . root in (-1, 0) and this is a contradiction � • Theorem 9. All natural powers of • Proof. Suppose that exist a chromatic polynomial Saeid Alikhani, UPM cannot be chromatic root. is a chromatic root, that is there 16
Such that . Therefore So we can say that so is a root of the polynomial, . Since have a root say. Since and so is a root of contradiction. � How about (2 n+1)- anacci? ? Saeid Alikhani, UPM and in (-1, 0). Therefore , we have a 17
• We think that (2 n + 1)-anacci numbers and all natural power of them also cannot be chromatic roots, but we are not able to prove it! • Theorem 10. (Dong et al [4]) Let polynomial of a graph G of order n. Then for any be a chromatic (where equality holds if and only if G is a tree). � • Theorem 11. For every natural n, can not be a root of chromatic polynomial of graph G with at most 4 n + 2 vertices. • Proof. We know that (2 n+1)- anacci is a root of Saeid Alikhani, UPM 18
• Now suppose that there exist a chromatic polynomial , such that, Therefore, there exist that • By Theorem 7, Saeid Alikhani, UPM , such . We have for , and so 19
• . have • So, we have By the above equalities, we . By Theorem 10, . Therefore P(x) cannot . be a chromatic polynomial, and this is a contradiction � Saeid Alikhani, UPM 20
• 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: Saeid Alikhani, UPM 21
• Question 1. (Beraha's question [1]) Is it true for every , there exists a plane triangulation G such that has a root in, , where is called the n-th Beraha constant (or number)? • Beraha et al. [3] proved that is an accumulation point of real chromatic roots of certain plane triangulations. • Jacobsen et al. [6] extended this to show that and are likewise accumulation points of real chromatic roots of plane triangulations. Saeid Alikhani, UPM 22
• 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]). Saeid Alikhani, UPM 23
• Now, let , where. • Here we ask the following question that is analogous to Beraha's question. • Question 2. Is it true that, for any , there exists a plane triangulation graph G such that has a root in ? ( ). � • Note that. 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). Saeid Alikhani, UPM 24
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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, B 7 and B 10 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. Saeid Alikhani, UPM 26
• [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 , 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. Saeid Alikhani, UPM 27
• [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, 2 nd 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-Step. Number. html (last accessed on Dec 2006) Saeid Alikhani, UPM 28
• Thanks for your attention! Saeid Alikhani, UPM 29
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