(Restrained) Chromatic Polynomials Aysel Erey Dalhousie University - - PowerPoint PPT Presentation

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 Restrained Chromatic Polynomials Roots of 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 Restrained Chromatic Polynomials Roots of 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 Restrained Chromatic Polynomials Roots of 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 Restrained Chromatic Polynomials Roots of 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 Restrained Chromatic Polynomials Roots of 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 Restrained Chromatic Polynomials Roots of 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 Restrained Chromatic Polynomials Roots of 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 Restrained Chromatic Polynomials Roots of 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 Restrained Chromatic Polynomials Roots of 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(Km,n) has a nonreal root z with ℜ(z) > m for all sufficiently large n.

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Roots of Chromatic Polynomials

The roots of π(Kn,n, x) for 2 ≤ n ≤ 40 in the z-plane

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Roots of Chromatic Polynomials

The roots of π(Kn,2n, x) for 2 ≤ n ≤ 25 in the z-plane

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Roots of Chromatic Polynomials

The roots of π(Kn,3n, x) for 2 ≤ n ≤ 20 in the z-plane

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Roots of 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 Restrained Chromatic Polynomials Roots of Chromatic Polynomials

Hermite-Biehler Let f (x) ∈ R[x] be standard, and write f (x) = f e(x2) + xf o(x2). Set t = x2. 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 Restrained Chromatic Polynomials Roots of 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 π(G, x) =

n

• i=1

(−1)n−ihi(G)xi, where hi(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 Restrained Chromatic Polynomials Roots of Chromatic Polynomials

Limit of chromatic roots of complete bipartite graphs

Definition For a sequence {fn(x)} of polynomials, z is called a limit of roots

• f {fn(x)} if there is a sequence {zn} such that fn(zn) = 0 and zn

converges to z.

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Roots of Chromatic Polynomials

Limit of chromatic roots of complete bipartite graphs

Definition For a sequence {fn(x)} of polynomials, z is called a limit of roots

• f {fn(x)} if there is a sequence {zn} such that fn(zn) = 0 and zn

converges to z. Theorem(J.B. and A.E.) Suppose that m ≥ 2 is fixed, then the set of limits of roots of π(Km,n, x) is exactly {z ∈ C | ℜ(z) = 1 + m 2 } ∪ {0, 1, . . . , ⌊m/2⌋}

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Roots of Chromatic Polynomials

π(Km,n, x) =

m

• k=1

S(m, k)(x)k(x − k)n where S(m, k) is the Stirling number of the second kind.

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Roots of Chromatic Polynomials

Beraha-Kahane-Weiss Suppose that {fn(x)} is a family of polynomials such that fn(x) = α1(x)λ1(x)n + α2(x)λ2(x)n · · · + αk(x)λk(x)n where αi(x) and λi(x) are fixed nonzero polynomials such that no pair i = j is αi(x) ≡ wαj(x) for some w ∈ C of unit modulus. Then the limits of roots of {fn(x)} are exactly those z satisfying two or more of the αi(z) are of equal modulus and strictly greater (in modulus) than the others; or for some j, αj(z) has modulus strictly greater than all the

• ther αi(z) have and αj(z) = 0.

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Roots of Chromatic Polynomials

The roots of π(K5,n, x) for 2 ≤ n ≤ 50 in the z-plane

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Roots of Chromatic Polynomials

Complete graph minus a matching Kn − qK2

π(Kn − qK2, x) =

q−1

• k=0

q − 1 k

• (x − n + 2 + k)(x)n−1−k

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Roots of Chromatic Polynomials

Complete graph minus a matching Kn − qK2

π(Kn − qK2, x) =

q−1

• k=0

q − 1 k

• (x − n + 2 + k)(x)n−1−k

Theorem (J.B-A.E 2013) Let G = Kn − qK2 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

• 2q. In particular, every

chromatic root z satisfies ℜ(z) < n − 2 + 3

2q and |ℑ(z)| < 3 2q.

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints Open Problems

Variants of graph colourings

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints 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

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints 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). Restrained Coloring Let each vertex v has a finite set r(v) of forbidden colors (r is called a restraint function), then a proper vertex coloring c is called a coloring permitted by r if for each vertex v, c(v) is not from r(v).

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints Open Problems

Restrained Chromatic Polynomials and Basic Properties

Definition The restrained chromatic polynomial πr(G, x) of G counts the number of x-colorings permitted by restraint r.

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints Open Problems

Restrained Chromatic Polynomials and Basic Properties

Definition The restrained chromatic polynomial πr(G, x) of G counts the number of x-colorings permitted by restraint r. πr(G, x) generalizes π(G, x) .

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints Open Problems

Restrained Chromatic Polynomials and Basic Properties

Definition The restrained chromatic polynomial πr(G, x) of G counts the number of x-colorings permitted by restraint r. πr(G, x) generalizes π(G, x) . Theorem (J.B.-A.E.-J.L.) πr(G, x) is a polynomial function of x for all sufficiently large x.

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints Open Problems

Restrained Chromatic Polynomials and Basic Properties

Definition The restrained chromatic polynomial πr(G, x) of G counts the number of x-colorings permitted by restraint r. πr(G, x) generalizes π(G, x) . Theorem (J.B.-A.E.-J.L.) πr(G, x) is a polynomial function of x for all sufficiently large x. πr(G, x) is monic of degree n with integer coefficients that alternate in sign.

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints Open Problems

Restrained Chromatic Polynomials and Basic Properties

Definition The restrained chromatic polynomial πr(G, x) of G counts the number of x-colorings permitted by restraint r. πr(G, x) generalizes π(G, x) . Theorem (J.B.-A.E.-J.L.) πr(G, x) is a polynomial function of x for all sufficiently large x. πr(G, x) is monic of degree n with integer coefficients that alternate in sign. The constant term of πr(G, x) need not be 0.

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints Open Problems

Definition r is called an m-restraint if |r(u)| ≤ m for every u.

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints Open Problems

Definition r is called an m-restraint if |r(u)| ≤ m for every u. r is called an standard m-restraint if |r(u)| = m for every u.

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints Open Problems

Definition r is called an m-restraint if |r(u)| ≤ m for every u. r is called an standard m-restraint if |r(u)| = m for every u. If m = 1 we use the word simple.

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints Open Problems

Question Given a graph G and x large enough, what standard simple restraints permit the largest/smallest number of colorings?

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints Open Problems

Restraints permitting the smallest number of colorings

Theorem (J.B.-A.E.-J.L. 2013) Let x be sufficiently large. Then πr(G, x) ≥ πrc(G, x) for every standard m-restraint r where rc is a constant restraint on V (G).

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints Open Problems

Restraints permitting the smallest number of colorings

Theorem (J.B.-A.E.-J.L. 2013) Let x be sufficiently large. Then πr(G, x) ≥ πrc(G, x) for every standard m-restraint r where rc is a constant restraint on V (G).

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints Open Problems

Restraints permitting the largest number of colorings for complete graphs

Theorem(J.B.-A.E.-J.L. 2013) Let r : {v1, v2, . . . , vn} − → [n] be any standard simple restraint on Kn , then for all x ≥ n, πr(Kn, x) ≤ πr′(Kn, x), where r ′(vi) = i for all i ≤ n.

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints Open Problems

r1 = [{1}, {1}, {1}, {1}, {1}], r2 = [{1}, {1}, {1}, {1}, {2}], r3 = [{1}, {1}, {1}, {2}, {2}], r4 = [{1}, {1}, {1}, {2}, {3}], r5 = [{1}, {1}, {2}, {3}, {4}], r6 = [{1}, {1}, {2}, {2}, {3}] and r7 = [{1}, {2}, {3}, {4}, {5}]

Figure : Restraint chromatic polynomials of K5 with respect to r1(Black), r2(Green), r3(Pink), r4(Yellow), r5(Grey), r6(Orange) and r7(Red).

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints Open Problems

Restraints permitting the largest number of colorings for trees

Definition Let V (G) = A ∪ B be the partition of a bipartite graph G. Then a restraint function is called an alternating restraint, denoted ralt, if ralt is constant on both A and B individually but ralt(A) = ralt(B).

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints Open Problems

Restraints permitting the largest number of colorings for trees

Definition Let V (G) = A ∪ B be the partition of a bipartite graph G. Then a restraint function is called an alternating restraint, denoted ralt, if ralt is constant on both A and B individually but ralt(A) = ralt(B). Theorem (J.B.-A.E.-J.L. 2013) Let T be a tree on n vertices and r : V (T) → [n] be a standard simple restraint other than the alternating restraint, then for k ≥ 3, πr(T, k) < πralt(T, k).

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints Open Problems

Restrained chromatic polynomials can distinguish some nonisomorphic graphs that chromatic polynomials cannot.

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints Open Problems

Restrained chromatic polynomials can distinguish some nonisomorphic graphs that chromatic polynomials cannot. If T is a tree of order n, then its chromatic polynomial π(T, k) is equal to k(k − 1)n−1.

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints Open Problems

Restrained chromatic polynomials can distinguish some nonisomorphic graphs that chromatic polynomials cannot. If T is a tree of order n, then its chromatic polynomial π(T, k) is equal to k(k − 1)n−1. If T = Pn or Sn then πralt(Sn, x) > πralt(T, x) > πralt(Pn, x) for all sufficiently large x.

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints Open Problems

Restrained chromatic polynomials can distinguish some nonisomorphic graphs that chromatic polynomials cannot. If T is a tree of order n, then its chromatic polynomial π(T, k) is equal to k(k − 1)n−1. If T = Pn or Sn then πralt(Sn, x) > πralt(T, x) > πralt(Pn, x) for all sufficiently large x.

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints Open Problems

Restraints permitting the largest number of colorings for bipartite graphs

Conjecture (J.B.-A.E.-J.L.) Let r : {v1, v2, . . . , vn} − → [n] be any standard simple restraint on a bipartite graph G and x large enough. Then, πr(G, x) ≤ πralt(G, x).

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints Open Problems

r1 = [{1}, {1}, {1}, {1}], r2 = [{1}, {1}, {1}, {2}], r3 = [{1}, {1}, {2}, {2}], r4 = [{1}, {2}, {1}, {2}], r5 = [{1}, {1}, {2}, {3}], r6 = [{1}, {2}, {1}, {3}] and r7 = [{1}, {2}, {3}, {4}].

Figure : Restraint chromatic polynomials of C4 with respect to r1 (Red), r2 (Green), r3 (Black), r4 (Blue), r5 (Brown), r6 (Gold) and r7 (Orange).

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints Open Problems

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints Open Problems

Restraints permitting the largest number of colorings for

• dd cycles

Conjecture (J.B.-A.E.-J.L.) Let r : {v1, v2, . . . , vn} − → [n] be any standard simple restraint on

• dd cycle Cn and x large enough. Then, πr(Cn, x) ≤ πr′(Cn, x),

where r ′(v1) = 3, r ′(v2i+1) = 1 and r ′(v2i) = 2 for all 1 ≤ i ≤ (n − 1)/2.

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints Open Problems

Conjectures for cycles and bipartite graphs are true for all such graphs of order at most 6.

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints Open Problems

Conjectures for cycles and bipartite graphs are true for all such graphs of order at most 6. For complete graphs and trees, restraint function which permits the largest number of colorings is indeed a minimal coloring of the graph.

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints Open Problems

Conjectures for cycles and bipartite graphs are true for all such graphs of order at most 6. For complete graphs and trees, restraint function which permits the largest number of colorings is indeed a minimal coloring of the graph. Is this true in general?

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints Open Problems

Conjectures for cycles and bipartite graphs are true for all such graphs of order at most 6. For complete graphs and trees, restraint function which permits the largest number of colorings is indeed a minimal coloring of the graph. Is this true in general? No.

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints Open Problems

v2 v3 v6 v1 v4 v5

The graph has chromatic number 3. Let r1 = [1, 2, 3, 1, 2, 4] and r2 = [1, 2, 3, 1, 2, 3] be two standard restraints on G, then πr1(G, x) − πr2(G, x) = (x − 3)2 > 0 for all x large enough.

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints Open Problems

Open Problem Is it true that for almost all graphs the restraint function which maximizes the restrained chromatic polynomial is a minimal colouring of the graph?

Aysel Erey

(Restrained) Chromatic Polynomials

Chromatic Polynomials Restrained Chromatic Polynomials Introduction to Restrained Chromatic Polynomials Extremal Restraints Open Problems

Thanks for listening!

Aysel Erey

(Restrained) Chromatic Polynomials