Complete Acyclic Colorings GGTW 2019, Ghent, Belgium attler 2 , - - PowerPoint PPT Presentation

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Complete Acyclic Colorings GGTW 2019, Ghent, Belgium attler 2 , - - PowerPoint PPT Presentation

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Complete Acyclic Colorings GGTW 2019, Ghent, Belgium attler 2 , Kolja Knauer 3 and Stefan Felsner 1 , Winfried Hochst Raphael Steiner 1 1 Technische Universit at Berlin 2 FernUniversit at in Hagen 3 Universit e Aix-Marseille 11-14

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Complete Acyclic Colorings

GGTW 2019, Ghent, Belgium Stefan Felsner1, Winfried Hochst¨ attler2, Kolja Knauer3 and Raphael Steiner1

1Technische Universit¨

at Berlin

2FernUniversit¨

at in Hagen

3Universit´

e Aix-Marseille

11-14 August 2019

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Arboreal and Acyclic Colorings

An arboreal coloring of a graph G is a partition of the vertex set into subsets inducing forests. It is complete if there is a cycle in the merge of any two color classes.

Raphael Steiner Complete Acyclic Colorings

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SLIDE 3

Complete Colorings Graph Operations Upper Bounds Lower Bounds

Arboreal and Acyclic Colorings

An arboreal coloring of a graph G is a partition of the vertex set into subsets inducing forests. It is complete if there is a cycle in the merge of any two color classes. Vertex arboricity va(G) Minimum number of colors in arboreal coloring va(G) = 2 A-vertex arboricity ava(G) Maximum number of colors in complete arboreal coloring ava(G) = 4

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Arboreal and Acyclic Colorings

An acyclic coloring of a digraph D is a partition of the vertex set into subsets inducing acyclic digraphs. It is complete if there is a directed cycle in the merge of any two color classes. Dichromatic number χ(D) Minimum number of colors in acyclic coloring

  • χ(D) = 2

Adichromatic number adi(D) Maximum number of colors in complete acyclic coloring adi(D) = 3

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Complete Bipartite Graphs

va(Kn,n) = 2

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Complete Bipartite Graphs

ava(Kn,n) = n

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Complete Bipartite Graphs

ava(Kn,n) = n

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Subgraphs

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Subgraphs

ava(G) = 2 ava(G ′) = 3

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Subgraphs

ava(G) = 2 ava(G ′) = 3 Lemma If G ′ is an induced subgraph of G, then ava(G ′) ≤ ava(G).

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Subgraphs

ava(G) = 2 ava(G ′) = 3 Lemma If G ′ is an induced subgraph of G, then ava(G ′) ≤ ava(G).

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Subgraphs

ava(G) = 2 ava(G ′) = 3 Lemma If G ′ is an induced subgraph of G, then ava(G ′) ≤ ava(G). ava(G ′) = 3

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Subgraphs

ava(G) = 2 ava(G ′) = 3 Lemma If G ′ is an induced subgraph of G, then ava(G ′) ≤ ava(G). ava(G ′) = 3

Raphael Steiner Complete Acyclic Colorings

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SLIDE 14

Complete Colorings Graph Operations Upper Bounds Lower Bounds

Subgraphs

ava(G) = 2 ava(G ′) = 3 Lemma If G ′ is an induced subgraph of G, then ava(G ′) ≤ ava(G). ava(G ′) = 3

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Subgraphs

ava(G) = 2 ava(G ′) = 3 Lemma If G ′ is an induced subgraph of G, then ava(G ′) ≤ ava(G). ava(G ′) = 3 ava(G) ≥ 3

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Subgraphs

ava(G) = 2 ava(G ′) = 3 Lemma If G ′ is an induced subgraph of G, then ava(G ′) ≤ ava(G). Lemma If D′ is an induced subdigraph of D, then adi(D′) ≤ adi(D).

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Induced Minors and Subdivisions

Lemma If e is a simple edge, then ava(G/e) ≤ ava(G).

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Induced Minors and Subdivisions

Lemma If e is a simple edge, then ava(G/e) ≤ ava(G).

e

Raphael Steiner Complete Acyclic Colorings

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SLIDE 19

Complete Colorings Graph Operations Upper Bounds Lower Bounds

Induced Minors and Subdivisions

Lemma If e is a simple edge, then ava(G/e) ≤ ava(G).

e

Raphael Steiner Complete Acyclic Colorings

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SLIDE 20

Complete Colorings Graph Operations Upper Bounds Lower Bounds

Induced Minors and Subdivisions

Lemma If e is a simple edge, then ava(G/e) ≤ ava(G).

e

Raphael Steiner Complete Acyclic Colorings

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SLIDE 21

Complete Colorings Graph Operations Upper Bounds Lower Bounds

Induced Minors and Subdivisions

Lemma If e is a simple edge, then ava(G/e) ≤ ava(G).

e

Raphael Steiner Complete Acyclic Colorings

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SLIDE 22

Complete Colorings Graph Operations Upper Bounds Lower Bounds

Induced Minors and Subdivisions

Lemma If e is a simple edge, then ava(G/e) ≤ ava(G).

e

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Induced Minors and Subdivisions

Corollary If H is an induced minor of G, then ava(H) ≤ ava(G). H G

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Relation to Feedback Vertex Sets

Definition A feedback vertex set of a graph (digraph) is a vertex set whose deletion yields a forest (acyclic digraph).

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Relation to Feedback Vertex Sets

Definition A feedback vertex set of a graph (digraph) is a vertex set whose deletion yields a forest (acyclic digraph). Proposition ava(G) ≤ fv(G) + 1 for any graph G. adi(D) ≤ fv(D) + 1 for any digraph D.

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Relation to Feedback Vertex Sets

Definition A feedback vertex set of a graph (digraph) is a vertex set whose deletion yields a forest (acyclic digraph). Proposition ava(G) ≤ fv(G) + 1 for any graph G. adi(D) ≤ fv(D) + 1 for any digraph D. Proof. In a complete arboreal/acyclic coloring, at most one colour class is disjoint from a feedback vertex set.

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Relations between the Parameters

Theorem (Felsner, Hochst¨ attler, Knauer, S. ’19) ∃ Multi-graphs with bounded ava and unbounded fv. ∃ Simple digraphs with bounded adi and unbounded fv. For simple graphs, there is f such that fv(G) ≤ f (ava(G)). For simple graphs, ava(G) ∼ maxD adi(D).

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Relations between the Parameters

Theorem (Felsner, Hochst¨ attler, Knauer, S. ’19) Let G be a non-trivial minor-closed class of simple graphs. There is f such that for D orientation of G ∈ G: fv(D) ≤ f (adi(D)). There is f (k) = O(k2 log k) such that for all G ∈ G: fv(G) ≤ f (ava(G)).

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Degeneracy vs. ava

Theorem There is f such that for all simple graphs G: deg(G) ≤ f (ava(G)).

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Degeneracy vs. ava

Theorem There is f such that for all simple graphs G: deg(G) ≤ f (ava(G)). Theorem (K¨ uhn and Osthus, 2004) For s ≥ 1 and every graph H there is d(s, H) ≥ 1 such that every G with δ(G) ≥ d(s, H) contains Ks,s as a subgraph or an induced subdivision of H.

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Degeneracy vs. ava

Theorem There is f such that for all simple graphs G: deg(G) ≤ f (ava(G)). Proof. If deg(G) ≥ d(s, Ks,s), then ava(G) ≥ s.

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

fv(G) ≤ f (ava(G))

Proof by contradiction: Assume ∃ sequence G1, G2, G3, . . . such that fv(Gi) → ∞ and ava(Gi) bounded.

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

fv(G) ≤ f (ava(G))

Proof by contradiction: Assume ∃ sequence G1, G2, G3, . . . such that fv(Gi) → ∞ and ava(Gi) bounded. Theorem (Erd˝

  • s and P´
  • sa ’65)

There is f (k) = O(k log k) such that for all graphs: cp(G) ≤ fv(G) ≤ f (cp(G)).

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

fv(G) ≤ f (ava(G))

Proof by contradiction: Assume ∃ sequence G1, G2, G3, . . . such that fv(Gi) → ∞ and ava(Gi) bounded. Theorem (Erd˝

  • s and P´
  • sa ’65)

There is f (k) = O(k log k) such that for all graphs: cp(G) ≤ fv(G) ≤ f (cp(G)). Therefore: cp(Gi) → ∞, and deg(Gi) ≤ d.

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

fv(G) ≤ f (ava(G))

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

fv(G) ≤ f (ava(G))

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

fv(G) ≤ f (ava(G))

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

fv(G) ≤ f (ava(G))

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

fv(G) ≤ f (ava(G))

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

fv(G) ≤ f (ava(G))

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

fv(G) ≤ f (ava(G))

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

fv(G) ≤ f (ava(G))

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

fv(G) ≤ f (ava(G))

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

fv(G) ≤ f (ava(G))

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

fv(G) ≤ f (ava(G))

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

The End.

Thank you.

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

fv(G) ≤ f (ava(G))

k: Number of short cycles. L: Maximum length of short cycle. d: Upper bound for degeneracy. N: Number of vertices in short cycles. k 2

  • ≤ Number of edges in induced subgraph ≤ dN ≤ dL · k

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Multigraphs

ava = 2, fv = n

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Simple digraphs

n = 3, k = 4

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Simple digraphs

adi ≤ k, fv = n

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Relationship of ava and adi

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Relationship of ava and adi adi(D) ≤ fv(D) + 1 ≤ fv(G) + 1 ≤ f (ava(G)) + 1.

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Relationship of ava and adi max

D adi(D) ≤ g(ava(G)).

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Relationship of ava and adi ava(G) ≤ h(max

D adi(D)))?

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Relationship of ava and adi

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Relationship of ava and adi

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Relationship of ava and adi

1 2 6 5 4 3

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Relationship of ava and adi

1 2 6 5 4 3

Raphael Steiner Complete Acyclic Colorings

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Complete Colorings Graph Operations Upper Bounds Lower Bounds

Relationship of ava and adi

1 2 6 5 4 3

Raphael Steiner Complete Acyclic Colorings