Chromatic Polynomials Restrained Chromatic Polynomials (Restrained) Chromatic Polynomials Aysel Erey Dalhousie University CanaDAM 2013, St. John’s June 13, 2013 Joint work with Jason Brown Aysel Erey (Restrained) Chromatic Polynomials
Chromatic Polynomials Roots of Chromatic Polynomials Restrained Chromatic Polynomials Definition The chromatic polynomial π ( G , k ) counts the number of (proper) k -colorings of G . Aysel Erey (Restrained) Chromatic Polynomials
Chromatic Polynomials Roots of Chromatic Polynomials Restrained Chromatic Polynomials Definition The chromatic polynomial π ( G , k ) counts the number of (proper) k -colorings of G . Theorem Let G be a connected graph of order n. The chromatic polynomial π ( G , k ) is a polynomial in k; Aysel Erey (Restrained) Chromatic Polynomials
Chromatic Polynomials Roots of Chromatic Polynomials Restrained Chromatic Polynomials Definition The chromatic polynomial π ( G , k ) counts the number of (proper) k -colorings of G . Theorem Let G be a connected graph of order n. The chromatic polynomial π ( G , k ) is a polynomial in k; has degree n; Aysel Erey (Restrained) Chromatic Polynomials
Chromatic Polynomials Roots of Chromatic Polynomials Restrained Chromatic Polynomials Definition The chromatic polynomial π ( G , k ) counts the number of (proper) k -colorings of G . Theorem Let G be a connected graph of order n. The chromatic polynomial π ( G , k ) is a polynomial in k; has degree n; has leading coefficient 1 ; Aysel Erey (Restrained) Chromatic Polynomials
Chromatic Polynomials Roots of Chromatic Polynomials Restrained Chromatic Polynomials Definition The chromatic polynomial π ( G , k ) counts the number of (proper) k -colorings of G . Theorem Let G be a connected graph of order n. The chromatic polynomial π ( G , k ) is a polynomial in k; has degree n; has leading coefficient 1 ; has integer coefficients with alternating signs and Aysel Erey (Restrained) Chromatic Polynomials
Chromatic Polynomials Roots of Chromatic Polynomials Restrained Chromatic Polynomials Definition The chromatic polynomial π ( G , k ) counts the number of (proper) k -colorings of G . Theorem Let G be a connected graph of order n. The chromatic polynomial π ( G , k ) is a polynomial in k; has degree n; has leading coefficient 1 ; has integer coefficients with alternating signs and the exponent of its smallest term is 1 . Aysel Erey (Restrained) Chromatic Polynomials
Chromatic Polynomials Roots of Chromatic Polynomials Restrained Chromatic Polynomials Resolution of a conjecture on chromatic roots Conjecture (Dong-Koh-Teo 2004) Let G be a graph of tree-with k ≥ 2 and z ∈ C be a chromatic root of G , then ℜ ( z ) ≤ k . Aysel Erey (Restrained) Chromatic Polynomials
Chromatic Polynomials Roots of Chromatic Polynomials Restrained Chromatic Polynomials Resolution of a conjecture on chromatic roots Conjecture (Dong-Koh-Teo 2004) Let G be a graph of tree-with k ≥ 2 and z ∈ C be a chromatic root of G , then ℜ ( z ) ≤ k . Theorem (J.B. and A.E. 2013) For any integer k ≥ 2 there exist infinitely many graphs with tree-width k and chromatic roots z such that ℜ ( z ) > k . Aysel Erey (Restrained) Chromatic Polynomials
Chromatic Polynomials Roots of Chromatic Polynomials Restrained Chromatic Polynomials Resolution of a conjecture on chromatic roots Conjecture (Dong-Koh-Teo 2004) Let G be a graph of tree-with k ≥ 2 and z ∈ C be a chromatic root of G , then ℜ ( z ) ≤ k . Theorem (J.B. and A.E. 2013) For any integer k ≥ 2 there exist infinitely many graphs with tree-width k and chromatic roots z such that ℜ ( z ) > k . Theorem (J.B. and A.E. 2013) Suppose that m ≥ 2 is fixed. π ( K m , n ) has a nonreal root z with ℜ ( z ) > m for all sufficiently large n . Aysel Erey (Restrained) Chromatic Polynomials
Chromatic Polynomials Roots of Chromatic Polynomials Restrained Chromatic Polynomials The roots of π ( K n , n , x ) for 2 ≤ n ≤ 40 in the z -plane Aysel Erey (Restrained) Chromatic Polynomials
Chromatic Polynomials Roots of Chromatic Polynomials Restrained Chromatic Polynomials The roots of π ( K n , 2 n , x ) for 2 ≤ n ≤ 25 in the z -plane Aysel Erey (Restrained) Chromatic Polynomials
Chromatic Polynomials Roots of Chromatic Polynomials Restrained Chromatic Polynomials The roots of π ( K n , 3 n , x ) for 2 ≤ n ≤ 20 in the z -plane Aysel Erey (Restrained) Chromatic Polynomials
Chromatic Polynomials Roots of Chromatic Polynomials Restrained Chromatic Polynomials C.Thomassen (1997) If G is a graph of tree-with k ≥ 2, then its real chromatic roots are bounded above by k . Aysel Erey (Restrained) Chromatic Polynomials
Chromatic Polynomials Roots of Chromatic Polynomials Restrained Chromatic Polynomials Hermite-Biehler Let f ( x ) ∈ R [ x ] be standard, and write f ( x ) = f e ( x 2 ) + xf o ( x 2 ). Set t = x 2 . Then f ( x ) is Hurwitz quasi-stable if and only if both f e ( t ) and f o ( t ) are standard, have only nonpositive zeros, and f o ( t ) ≺ f e ( t ). Aysel Erey (Restrained) Chromatic Polynomials
Chromatic Polynomials Roots of Chromatic Polynomials Restrained Chromatic Polynomials Whitney’s Broken-cycle Theorem Let G be a graph of order n and size m , and let β : E ( G ) → { 1 , 2 , . . . , m } be any bijection. Then n � ( − 1) n − i h i ( G ) x i , π ( G , x ) = i =1 where h i ( G ) is the number of spanning subgraphs of G that have exactly n − i edges and that contain no broken cycles with respect to β . Aysel Erey (Restrained) Chromatic Polynomials
Chromatic Polynomials Roots of Chromatic Polynomials Restrained Chromatic Polynomials Limit of chromatic roots of complete bipartite graphs Definition For a sequence { f n ( x ) } of polynomials, z is called a limit of roots of { f n ( x ) } if there is a sequence { z n } such that f n ( z n ) = 0 and z n converges to z . Aysel Erey (Restrained) Chromatic Polynomials
Chromatic Polynomials Roots of Chromatic Polynomials Restrained Chromatic Polynomials Limit of chromatic roots of complete bipartite graphs Definition For a sequence { f n ( x ) } of polynomials, z is called a limit of roots of { f n ( x ) } if there is a sequence { z n } such that f n ( z n ) = 0 and z n converges to z . Theorem(J.B. and A.E.) Suppose that m ≥ 2 is fixed, then the set of limits of roots of π ( K m , n , x ) is exactly { z ∈ C | ℜ ( z ) = 1 + m } ∪ { 0 , 1 , . . . , ⌊ m / 2 ⌋} 2 Aysel Erey (Restrained) Chromatic Polynomials
Chromatic Polynomials Roots of Chromatic Polynomials Restrained Chromatic Polynomials m � S ( m , k )( x ) k ( x − k ) n π ( K m , n , x ) = k =1 where S ( m , k ) is the Stirling number of the second kind. Aysel Erey (Restrained) Chromatic Polynomials
Chromatic Polynomials Roots of Chromatic Polynomials Restrained Chromatic Polynomials Beraha-Kahane-Weiss Suppose that { f n ( x ) } is a family of polynomials such that f n ( x ) = α 1 ( x ) λ 1 ( x ) n + α 2 ( x ) λ 2 ( x ) n · · · + α k ( x ) λ k ( x ) n where α i ( x ) and λ i ( x ) are fixed nonzero polynomials such that no pair i � = j is α i ( x ) ≡ w α j ( x ) for some w ∈ C of unit modulus. Then the limits of roots of { f n ( x ) } are exactly those z satisfying two or more of the α i ( z ) are of equal modulus and strictly greater (in modulus) than the others; or for some j , α j ( z ) has modulus strictly greater than all the other α i ( z ) have and α j ( z ) = 0. Aysel Erey (Restrained) Chromatic Polynomials
Chromatic Polynomials Roots of Chromatic Polynomials Restrained Chromatic Polynomials The roots of π ( K 5 , n , x ) for 2 ≤ n ≤ 50 in the z -plane Aysel Erey (Restrained) Chromatic Polynomials
Chromatic Polynomials Roots of Chromatic Polynomials Restrained Chromatic Polynomials Complete graph minus a matching K n − qK 2 q − 1 � q − 1 � � π ( K n − qK 2 , x ) = ( x − n + 2 + k )( x ) n − 1 − k k k =0 Aysel Erey (Restrained) Chromatic Polynomials
Chromatic Polynomials Roots of Chromatic Polynomials Restrained Chromatic Polynomials Complete graph minus a matching K n − qK 2 q − 1 � q − 1 � � π ( K n − qK 2 , x ) = ( x − n + 2 + k )( x ) n − 1 − k k k =0 Theorem (J.B-A.E 2013) Let G = K n − qK 2 with q > 0, then every noninteger chromatic root of G lies in the union U of the discs centered at n − q − 1 , n − q , . . . , n − 2 each of radius 3 2 q . In particular, every chromatic root z satisfies ℜ ( z ) < n − 2 + 3 2 q and |ℑ ( z ) | < 3 2 q . Aysel Erey (Restrained) Chromatic Polynomials
Introduction to Restrained Chromatic Polynomials Chromatic Polynomials Extremal Restraints Restrained Chromatic Polynomials Open Problems Variants of graph colourings Aysel Erey (Restrained) Chromatic Polynomials
Introduction to Restrained Chromatic Polynomials Chromatic Polynomials Extremal Restraints Restrained Chromatic Polynomials Open Problems Variants of graph colourings List Coloring Let each vertex v has a finite set L ( v ) of colors for use, then a proper vertex coloring c is called is a list coloring of G if for each vertex v , c ( v ) is from L ( v ). Aysel Erey (Restrained) Chromatic Polynomials
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