The NL-coflow polynomial (joint work with W. Hochstttler) MCW 2019 - - PowerPoint PPT Presentation

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The NL-coflow polynomial (joint work with W. Hochstttler) MCW 2019 - - PowerPoint PPT Presentation

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Discrete Mathematics and Optimization Johanna Wiehe: The NL-coflow polynomial (joint work with W. Hochstttler) MCW 2019 The NL-coflow polynomial Johanna Wiehe Discrete Mathematics and Optimization motivation undirected graph G = ( V , E )

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SLIDE 1

Discrete Mathematics and Optimization

Johanna Wiehe:

The NL-coflow polynomial

(joint work with W. Hochstättler)

MCW 2019 The NL-coflow polynomial Johanna Wiehe

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SLIDE 2

Discrete Mathematics and Optimization

motivation

undirected graph G = (V, E) chromatic polynomial P(G, k) = # proper k-colorings

MCW 2019 The NL-coflow polynomial Johanna Wiehe

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SLIDE 3

Discrete Mathematics and Optimization

motivation

undirected graph G = (V, E) chromatic polynomial P(G, k) = # proper k-colorings flow polynomial Φ(G, k) = # nowhere-zero k-flows

MCW 2019 The NL-coflow polynomial Johanna Wiehe

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SLIDE 4

Discrete Mathematics and Optimization

motivation

undirected graph G = (V, E) chromatic polynomial P(G, k) = # proper k-colorings flow polynomial Φ(G, k) = # nowhere-zero k-flows connection: G planar, without bridges: G is k-colorable ⇐ ⇒ G has dual NZ-k-flow

MCW 2019 The NL-coflow polynomial Johanna Wiehe

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SLIDE 5

Discrete Mathematics and Optimization

motivation

undirected graph G = (V, E) chromatic polynomial P(G, k) = # proper k-colorings flow polynomial Φ(G, k) = # nowhere-zero k-flows connection: G planar, without bridges: G is k-colorable ⇐ ⇒ G has dual NZ-k-flow generalization: Tutte Polynomial TG(x, y) =

  • S⊆E

(x − 1)rk(E)−rk(S)(y − 1)|S|−rk(S)

MCW 2019 The NL-coflow polynomial Johanna Wiehe

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SLIDE 6

Discrete Mathematics and Optimization

motivation

a proper coloring with 3 colors

MCW 2019 The NL-coflow polynomial Johanna Wiehe

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Discrete Mathematics and Optimization

motivation

1 2 1 1 −1 1 1 1 1 −2 1 2 a proper coloring with 3 colors an NZ-3-flow

MCW 2019 The NL-coflow polynomial Johanna Wiehe

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SLIDE 8

Discrete Mathematics and Optimization

motivation

directed graph D = (V, A) dichromatic number

→

χ(D) = minimal number of acyclic colorings

MCW 2019 The NL-coflow polynomial Johanna Wiehe

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SLIDE 9

Discrete Mathematics and Optimization

motivation

directed graph D = (V, A) dichromatic number

→

χ(D) = minimal number of acyclic colorings Neumann-Lara-coflow polynomial ψD

NL(k) = # NL-k-coflows = # acyclic colorings with k colors

MCW 2019 The NL-coflow polynomial Johanna Wiehe

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SLIDE 10

Discrete Mathematics and Optimization

motivation

directed graph D = (V, A) dichromatic number

→

χ(D) = minimal number of acyclic colorings Neumann-Lara-coflow polynomial ψD

NL(k) = # NL-k-coflows = # acyclic colorings with k colors

connection: D without loops:

→

χ(D) ≤ k ⇐ ⇒ D has an NL-k-coflow

MCW 2019 The NL-coflow polynomial Johanna Wiehe

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SLIDE 11

Discrete Mathematics and Optimization

motivation

directed graph D = (V, A) dichromatic number

→

χ(D) = minimal number of acyclic colorings Neumann-Lara-coflow polynomial ψD

NL(k) = # NL-k-coflows = # acyclic colorings with k colors

connection: D without loops:

→

χ(D) ≤ k ⇐ ⇒ D has an NL-k-coflow conjecture (Neumann-Lara) every orientation of a simple planar graph can be acyclically colored with two colors

MCW 2019 The NL-coflow polynomial Johanna Wiehe

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SLIDE 12

Discrete Mathematics and Optimization

motivation

an acyclic coloring with 2 colors

MCW 2019 The NL-coflow polynomial Johanna Wiehe

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Discrete Mathematics and Optimization

motivation

−1 1 1 1 −1 −1 an acyclic coloring with 2 colors an NL-2-coflow

MCW 2019 The NL-coflow polynomial Johanna Wiehe

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Discrete Mathematics and Optimization

NL-Coflows

Definition (Hochstättler) Let D = (V, A) be a digraph. A coflow is a map f, that satisfies Kirchhoff’s law

  • f flow conservation for (weak) cycles
  • a∈C+

f(a) =

  • a∈C−

f(a)

MCW 2019 The NL-coflow polynomial Johanna Wiehe

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SLIDE 15

Discrete Mathematics and Optimization

NL-Coflows

Definition (Hochstättler) Let D = (V, A) be a digraph. A coflow is a map f, that satisfies Kirchhoff’s law

  • f flow conservation for (weak) cycles
  • a∈C+

f(a) =

  • a∈C−

f(a) Let G be a finite Abelian group. An NL-G-coflow in D is a coflow f : A − → G, such that supp(f) contains a feedback arc set (i.e. S ⊆ A s.t. D − S is acyclic).

MCW 2019 The NL-coflow polynomial Johanna Wiehe

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SLIDE 16

Discrete Mathematics and Optimization

NL-Coflows

Definition (Hochstättler) Let D = (V, A) be a digraph. A coflow is a map f, that satisfies Kirchhoff’s law

  • f flow conservation for (weak) cycles
  • a∈C+

f(a) =

  • a∈C−

f(a) Let G be a finite Abelian group. An NL-G-coflow in D is a coflow f : A − → G, such that supp(f) contains a feedback arc set (i.e. S ⊆ A s.t. D − S is acyclic). For k ≥ 2, a coflow f : A − → {0, ±1, ..., ±(k − 1)} is an NL-k-coflow, if supp(f) is contains a feedback arc set.

MCW 2019 The NL-coflow polynomial Johanna Wiehe

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Discrete Mathematics and Optimization

Möbius inversion

Definition Let (P, ≤) be a finite poset, then the Möbius function is defined as follows µ : P × P → Z, µ(x, y) :=      , if x y 1 , if x = y −

x≤z<y µ(x, z)

, if x < y. Theorem (inversion from above) Let (P, ≤) be a finite poset, f, g : P → K functions and µ the Möbius

  • function. Then the following equivalence holds

f(x) =

  • y≥x

g(y), for all x ∈ P ⇐ ⇒ g(x) =

  • y≥x

µ(x, y)f(y), for all x ∈ P.

MCW 2019 The NL-coflow polynomial Johanna Wiehe

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Discrete Mathematics and Optimization

Let D = (V, A) be a digraph and let fk : 2A → Z count all G-coflows and let gk : 2A → Z count all NL-G-coflows. Using C :=

  • A/C | ∃ C1, ..., Cr directed cycles, such that C =

r

  • i=1

Ci

  • with “ ⊇ “

we find fk(A) =

  • B∈C

gk(B),

MCW 2019 The NL-coflow polynomial Johanna Wiehe

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Discrete Mathematics and Optimization

Let D = (V, A) be a digraph and let fk : 2A → Z count all G-coflows and let gk : 2A → Z count all NL-G-coflows. Using C :=

  • A/C | ∃ C1, ..., Cr directed cycles, such that C =

r

  • i=1

Ci

  • with “ ⊇ “

we find fk(A) =

  • B∈C

gk(B), that is

ψD

NL(k) = gk(A) = B∈C

µ(A, B) · krk(B).

MCW 2019 The NL-coflow polynomial Johanna Wiehe

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Discrete Mathematics and Optimization

another representation

Consider the poset P = {B ⊆ A | D[B] is totally cyclic subdigraph of D},

MCW 2019 The NL-coflow polynomial Johanna Wiehe

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Discrete Mathematics and Optimization

another representation

Consider the poset P = {B ⊆ A | D[B] is totally cyclic subdigraph of D}, which can be represented by a polyhedral cone Mx ≤ 0

MCW 2019 The NL-coflow polynomial Johanna Wiehe

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Discrete Mathematics and Optimization

another representation

Consider the poset P = {B ⊆ A | D[B] is totally cyclic subdigraph of D}, which can be represented by a polyhedral cone Mx ≤ 0 we obtain

⇒ ψD

NL(k) = B∈P

(−1)rkP(B)krk(A/B).

MCW 2019 The NL-coflow polynomial Johanna Wiehe

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Discrete Mathematics and Optimization

Symmetric digraphs

Let D = (V, A) be a symmetric digraph and G = (V, E) its underlying undirected

  • graph. Then we have

ψD

NL(x) = P(G, x) · x−c(G).

MCW 2019 The NL-coflow polynomial Johanna Wiehe

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Discrete Mathematics and Optimization

Symmetric digraphs

Let D = (V, A) be a symmetric digraph and G = (V, E) its underlying undirected

  • graph. Then we have

ψD

NL(x) = P(G, x) · x−c(G).

  • Proof. The polyhedron described by

(M, −M) ⇀ x

↼

x

  • = 0

⇀

xi +

↼

xi ≥ 1 ∀i

⇀

x ,

↼

x ≥ 0 is unbounded, thus the face poset is contractible!

MCW 2019 The NL-coflow polynomial Johanna Wiehe

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Discrete Mathematics and Optimization

  • pen problems

How does the NL-coflow polynomial (= # acyclic colorings) of totally cyclic digraphs look like? In general, how does the NL-coflow polynomial of complete digraphs look like? Is there a meaningful two variable polynomial combining the dichromatic and the NL-flow polynomial as the Tutte polynomial does in the classical case? How many vertices suffice to create a 5-chromatic tournament?

(a 3-chromatic tournament has at least 7 vertices, a 4-chromatic tournament at least 11)

...

MCW 2019 The NL-coflow polynomial Johanna Wiehe

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Discrete Mathematics and Optimization

Altenbokum, B.: Algebraische NL-Flüsse und Polynome. Master’s thesis, FernUniversität in Hagen, 2018. Hochstättler, W.: A flow theory for the dichromatic number. In: European J. Combinatorics, 66, 2017. Hochstättler, W., Wiehe, J.: The NL-flow polynomial. Tech.Rep.feU-dmo053.18, FernUniversität in Hagen, 2018. Neumann-Lara, V.: The dichromatic number of a digraph. In: J. of Combin. Theory, Series B, 33, 1982. Tutte, W.T.: A contribution to the theory of chromatic polynomials. Canad. J. Math., 6, 1954. Thank you for your attention.

MCW 2019 The NL-coflow polynomial Johanna Wiehe