Discrete Mathematics and Optimization Johanna Wiehe: The NL-coflow polynomial (joint work with W. Hochstättler) MCW 2019 The NL-coflow polynomial Johanna Wiehe
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
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
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
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 � ( x − 1 ) rk ( E ) − rk ( S ) ( y − 1 ) | S |− rk ( S ) T G ( x , y ) = S ⊆ E MCW 2019 The NL-coflow polynomial Johanna Wiehe
Discrete Mathematics and Optimization motivation a proper coloring with 3 colors MCW 2019 The NL-coflow polynomial Johanna Wiehe
Discrete Mathematics and Optimization motivation 1 − 1 1 1 2 1 1 1 1 − 2 2 1 a proper coloring with 3 colors an NZ-3-flow MCW 2019 The NL-coflow polynomial Johanna Wiehe
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
Discrete Mathematics and Optimization motivation directed graph D = ( V , A ) → dichromatic number χ ( D ) = minimal number of acyclic colorings Neumann-Lara-coflow polynomial ψ D NL ( k ) = # NL- k -coflows = # acyclic colorings with k colors MCW 2019 The NL-coflow polynomial Johanna Wiehe
Discrete Mathematics and Optimization motivation directed graph D = ( V , A ) → dichromatic number χ ( D ) = minimal number of acyclic colorings Neumann-Lara-coflow polynomial ψ D 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
Discrete Mathematics and Optimization motivation directed graph D = ( V , A ) → dichromatic number χ ( D ) = minimal number of acyclic colorings Neumann-Lara-coflow polynomial ψ D 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
Discrete Mathematics and Optimization motivation an acyclic coloring with 2 colors MCW 2019 The NL-coflow polynomial Johanna Wiehe
Discrete Mathematics and Optimization motivation − 1 − 1 1 1 1 0 0 − 1 0 an acyclic coloring with 2 colors an NL-2-coflow MCW 2019 The NL-coflow polynomial Johanna Wiehe
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 of flow conservation for (weak) cycles � � f ( a ) = f ( a ) a ∈ C + a ∈ C − MCW 2019 The NL-coflow polynomial Johanna Wiehe
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 of flow conservation for (weak) cycles � � f ( a ) = f ( a ) a ∈ C + a ∈ C − 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
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 of flow conservation for (weak) cycles � � f ( a ) = f ( a ) a ∈ C + a ∈ C − 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
Discrete Mathematics and Optimization Möbius inversion Definition Let ( P , ≤ ) be a finite poset, then the Möbius function is defined as follows , if x � y 0 µ : P × P → Z , µ ( 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 ) = g ( y ) , for all x ∈ P ⇐ ⇒ g ( x ) = µ ( x , y ) f ( y ) , for all x ∈ P . y ≥ x y ≥ x MCW 2019 The NL-coflow polynomial Johanna Wiehe
Discrete Mathematics and Optimization Let D = ( V , A ) be a digraph and let f k : 2 A → Z count all G -coflows and let g k : 2 A → Z count all NL- G -coflows. Using r � C := � A / C | ∃ C 1 , ..., C r directed cycles, such that C = � with “ ⊇ “ C i i = 1 we find � f k ( A ) = g k ( B ) , B ∈C MCW 2019 The NL-coflow polynomial Johanna Wiehe
Discrete Mathematics and Optimization Let D = ( V , A ) be a digraph and let f k : 2 A → Z count all G -coflows and let g k : 2 A → Z count all NL- G -coflows. Using r � C := � A / C | ∃ C 1 , ..., C r directed cycles, such that C = � with “ ⊇ “ C i i = 1 we find � f k ( A ) = g k ( B ) , B ∈C that is ψ D µ ( A , B ) · k rk ( B ) . NL ( k ) = g k ( A ) = � B ∈C MCW 2019 The NL-coflow polynomial Johanna Wiehe
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
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
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 ⇒ ψ D ( − 1 ) rk P ( B ) k rk ( A / B ) . NL ( k ) = � B ∈P MCW 2019 The NL-coflow polynomial Johanna Wiehe
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 ψ D NL ( x ) = P ( G , x ) · x − c ( G ) . MCW 2019 The NL-coflow polynomial Johanna Wiehe
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 ψ D NL ( x ) = P ( G , x ) · x − c ( G ) . Proof. The polyhedron described by � ⇀ � x ( M , − M ) = 0 ↼ x ⇀ ↼ x i + x i ≥ 1 ∀ i ⇀ ↼ x , x ≥ 0 is unbounded, thus the face poset is contractible! MCW 2019 The NL-coflow polynomial Johanna Wiehe
Discrete Mathematics and Optimization open 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
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
Recommend
More recommend
