Domination in tournaments Nicolas Bousquet Birmingham, June 2017 - - PowerPoint PPT Presentation

Domination in tournaments

Nicolas Bousquet Birmingham, June 2017

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Tournaments

A tournament is a digraph where either x → y or x ← y for every pair of vertices x, y.

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Tournaments

A tournament is a digraph where either x → y or x ← y for every pair of vertices x, y. A dominating set X ⊆ V is a set such that N+[X] = V .

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Tournaments

A tournament is a digraph where either x → y or x ← y for every pair of vertices x, y. A dominating set X ⊆ V is a set such that N+[X] = V . Notations : N−(v) : in-neighborhood of v excluding v. N−[v] : in-neighborhood of v including v. N+[v] : out-neighborhood of v including v.

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Domination in tournaments

There exist tournaments with domination number Ω(log n). Theorem

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Domination in tournaments

There exist tournaments with domination number Ω(log n). Theorem Question : What if add structure ? Example : Transitive tournaments have domination number 1 !

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Domination in tournaments

There exist tournaments with domination number Ω(log n). Theorem Question : What if add structure ? Example : Transitive tournaments have domination number 1 ! During the presentation today :

• k-majority tournaments. (Alon et al. ’04)
• Union of k partial orders. (B., Lochet, Thomass´

e ’17)

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k-majority tournaments

V = {1, ..., n}. Let ≺1, . . . , ≺2k−1 be total orders on V . The tournament realized by ≺1, . . . , ≺2k−1 has an arc i → j iff i ≻ j in at least k orders.

1 ≻ 2 ≻ 3 2 ≻ 3 ≻ 1 3 ≻ 1 ≻ 2

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k-majority tournaments

V = {1, ..., n}. Let ≺1, . . . , ≺2k−1 be total orders on V . The tournament realized by ≺1, . . . , ≺2k−1 has an arc i → j iff i ≻ j in at least k orders.

1 ≻ 2 ≻ 3 2 ≻ 3 ≻ 1 3 ≻ 1 ≻ 2 1 2 3

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k-majority tournaments

V = {1, ..., n}. Let ≺1, . . . , ≺2k−1 be total orders on V . The tournament realized by ≺1, . . . , ≺2k−1 has an arc i → j iff i ≻ j in at least k orders.

1 ≻ 2 ≻ 3 2 ≻ 3 ≻ 1 3 ≻ 1 ≻ 2 1 2 3

Tournament realized by 2k − 1 total orders. Definition (k-majority tournament) Example : 1-majority tournaments are transitive tournaments.

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k-majority tournaments

V = {1, ..., n}. Let ≺1, . . . , ≺2k−1 be total orders on V . The tournament realized by ≺1, . . . , ≺2k−1 has an arc i → j iff i ≻ j in at least k orders.

1 ≻ 2 ≻ 3 2 ≻ 3 ≻ 1 3 ≻ 1 ≻ 2 1 2 3

Tournament realized by 2k − 1 total orders. Definition (k-majority tournament) Example : 1-majority tournaments are transitive tournaments. Every k-majority tournament has a dominating set of size O(k · log(k)). Theorem (Alon, Brightwell, Kierstead, Kostochka, Winkler ’04)

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Outline of the proof

Every k-majority tournament has a dominating set of size O(k · log(k)). Theorem (Alon, Brightwell, Kierstead, Kostochka, Winkler ’04) A tournament of “bounded VC-dimension” has a do- minating set of bounded size.

1) Define VC-dimension for graphs. 2) Apply a result of Haussler and Welzl that implies that there is a dominating set of bounded size.

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Outline of the proof

Every k-majority tournament has a dominating set of size O(k · log(k)). Theorem (Alon, Brightwell, Kierstead, Kostochka, Winkler ’04) A tournament of “bounded VC-dimension” has a do- minating set of bounded size.

1) Define VC-dimension for graphs. 2) Apply a result of Haussler and Welzl that implies that there is a dominating set of bounded size.

Show that k-majority tournaments have bounded VC- dimension.

“Double counting” on the possible traces of neighborhoods on a set of vertices.

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Definitions

A hypergraph is a pair (V , E) where V is a set of vertices and E is a set of hyperedges (subsets of vertices).

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Definitions

A hypergraph is a pair (V , E) where V is a set of vertices and E is a set of hyperedges (subsets of vertices). A hitting set is a subset of vertices intersecting all the hyperedges. Transversality τ : minimum size of a hitting set.

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Definitions

A hypergraph is a pair (V , E) where V is a set of vertices and E is a set of hyperedges (subsets of vertices). A hitting set is a subset of vertices intersecting all the hyperedges. Transversality τ : minimum size of a hitting set. A fractional hitting set is a weight function on V such that w(X) ≥ 1 for any hyperedge X. Fractional transversality τ ∗ : minimum weight of w(V ).

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Definitions

A hypergraph is a pair (V , E) where V is a set of vertices and E is a set of hyperedges (subsets of vertices). A hitting set is a subset of vertices intersecting all the hyperedges. Transversality τ : minimum size of a hitting set. A fractional hitting set is a weight function on V such that w(X) ≥ 1 for any hyperedge X. Fractional transversality τ ∗ : minimum weight of w(V ). Remark : τ ≥ τ ∗.

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Definitions

A hypergraph is a pair (V , E) where V is a set of vertices and E is a set of hyperedges (subsets of vertices). A hitting set is a subset of vertices intersecting all the hyperedges. Transversality τ : minimum size of a hitting set. A fractional hitting set is a weight function on V such that w(X) ≥ 1 for any hyperedge X. Fractional transversality τ ∗ : minimum weight of w(V ). Remark : τ ≥ τ ∗. No converse function in general !

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VC-dimension

A set Y ⊆ X is a trace on X if there exists e ∈ E such that e ∩ X = Y . A set X ⊆ V is shattered iff all the traces

• n X exist.

The VC-dimension of a hypergraph is the maximum size of a shattered set.

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VC-dimension

A set Y ⊆ X is a trace on X if there exists e ∈ E such that e ∩ X = Y . A set X ⊆ V is shattered iff all the traces

• n X exist.

The VC-dimension of a hypergraph is the maximum size of a shattered set. Every hypergraph H of VC-dimension d satisfies τ ≤ 2dτ ∗log(11τ ∗). Theorem (Haussler, Welzl ’73)

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Transformation into hypergraphs

• Consider the hypergraph H where the hyperedges are the

closed in-neighborhoods of the vertices of T.

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Transformation into hypergraphs

• Consider the hypergraph H where the hyperedges are the

closed in-neighborhoods of the vertices of T.

• A hitting set of H is a dominating set of T.

Let X be a hitting set. For every v, N−[v] is a hyperedge. ⇒ For every v, X intersects N−[v]. ⇒ N+[X] = V .

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Transformation into hypergraphs

• Consider the hypergraph H where the hyperedges are the

closed in-neighborhoods of the vertices of T.

• A hitting set of H is a dominating set of T.

Let X be a hitting set. For every v, N−[v] is a hyperedge. ⇒ For every v, X intersects N−[v]. ⇒ N+[X] = V .

• A fractional hitting set of H is a weight function such that

w−[v] ≥ 1 for every v ∈ V .

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Transformation into hypergraphs

• Consider the hypergraph H where the hyperedges are the

closed in-neighborhoods of the vertices of T.

• A hitting set of H is a dominating set of T.

Let X be a hitting set. For every v, N−[v] is a hyperedge. ⇒ For every v, X intersects N−[v]. ⇒ N+[X] = V .

• A fractional hitting set of H is a weight function such that

w−[v] ≥ 1 for every v ∈ V .

• A shattered set X is a set of vertices such that for every

Y ⊆ X there exists v such that N+[v] ∩ X = Y .

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VC-dimension and domination

Every hypergraph H of VC-dimension d satisfies : τ ≤ 2dτ ∗log(11τ ∗). Theorem (Haussler, Welzl ’72)

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VC-dimension and domination

Every hypergraph H of VC-dimension d satisfies : τ ≤ 2dτ ∗log(11τ ∗). Theorem (Haussler, Welzl ’72) For any directed graph, there exists w : V → R+ such that :

• w(V ) = 2.
• For every v ∈ V , w(N−[v]) ≥ w(N+(v)).

Theorem (Fisher) Translation for tournaments : There is a weight function such that w(N−[v]) ≥ 1 and w(V ) = 2.

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VC-dimension and domination

Every hypergraph H of VC-dimension d satisfies : τ ≤ 2dτ ∗log(11τ ∗). Theorem (Haussler, Welzl ’72) For any directed graph, there exists w : V → R+ such that :

• w(V ) = 2.
• For every v ∈ V , w(N−[v]) ≥ w(N+(v)).

Theorem (Fisher) Translation for tournaments : There is a weight function such that w(N−[v]) ≥ 1 and w(V ) = 2. ⇒ τ ∗ ≤ 2.

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VC-dimension and domination

Every hypergraph H of VC-dimension d satisfies : τ ≤ 2dτ ∗log(11τ ∗). Theorem (Haussler, Welzl ’72) For any directed graph, there exists w : V → R+ such that :

• w(V ) = 2.
• For every v ∈ V , w(N−[v]) ≥ w(N+(v)).

Theorem (Fisher) Translation for tournaments : There is a weight function such that w(N−[v]) ≥ 1 and w(V ) = 2. ⇒ τ ∗ ≤ 2. In a tournament : Bounded VC-dimension ⇒ Bounded domination number.

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VC-dimension of k-majority tournaments

≺1 ≺2 ≺3 x1 x2 x3 x2 x1 x3 x3 x1 x2 a b c c a b a b c

Shattered set : X = {x1, x2, · · · , xℓ}. Given X, ≺i partitions V into |X| + 1 classes :

vertices before the 1st vertex of X, between the 1st and the second, ...

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VC-dimension of k-majority tournaments

≺1 ≺2 ≺3 x1 x2 x3 x2 x1 x3 x3 x1 x2 a b c c a b a b c

Shattered set : X = {x1, x2, · · · , xℓ}. Given X, ≺i partitions V into |X| + 1 classes :

vertices before the 1st vertex of X, between the 1st and the second, ...

⇒ ≺1, . . . , ≺2k−1 partition V into (|X| + 1)2k−1 classes.

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VC-dimension of k-majority tournaments

≺1 ≺2 ≺3 x1 x2 x3 x2 x1 x3 x3 x1 x2 a b c c a b a b c

Shattered set : X = {x1, x2, · · · , xℓ}. Given X, ≺i partitions V into |X| + 1 classes :

vertices before the 1st vertex of X, between the 1st and the second, ...

⇒ ≺1, . . . , ≺2k−1 partition V into (|X| + 1)2k−1 classes. Vertices in the same class have the same neighborhood in X. Claim Proof : u, v in the same class ⇒ ∀x, u ≺i x ⇔ v ≺i x.

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VC-dimension of k-majority tournaments

≺1 ≺2 ≺3 x1 x2 x3 x2 x1 x3 x3 x1 x2 a b c c a b a b c

Shattered set : X = {x1, x2, · · · , xℓ}. Given X, ≺i partitions V into |X| + 1 classes :

vertices before the 1st vertex of X, between the 1st and the second, ...

⇒ ≺1, . . . , ≺2k−1 partition V into (|X| + 1)2k−1 classes. Vertices in the same class have the same neighborhood in X. Claim Proof : u, v in the same class ⇒ ∀x, u ≺i x ⇔ v ≺i x. ⇒ There are at most (|X| + 1)2k−1 traces on X in the hypergraph.

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Upper bound on the size of X

Let X be a shattered set. X is shattered ⇒ 2|X| traces on X. k-majority tournament ⇒ At most (|X| + 1)2k−1 traces on X.

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Upper bound on the size of X

Let X be a shattered set. X is shattered ⇒ 2|X| traces on X. k-majority tournament ⇒ At most (|X| + 1)2k−1 traces on X. ⇒ 2|X| ≥ (|X| + 1)2k−1 ⇒ |X| ≤ O(k · log k)

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Upper bound on the size of X

Let X be a shattered set. X is shattered ⇒ 2|X| traces on X. k-majority tournament ⇒ At most (|X| + 1)2k−1 traces on X. ⇒ 2|X| ≥ (|X| + 1)2k−1 ⇒ |X| ≤ O(k · log k) VC-dimension : O(k · log k). Fractional transversality : ≤ 2. ⇒ τ ≤ O(k log k)

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Generalization

A set of k partial orders ≺1, ..., ≺k partition a tournament D if xi → xj ⇔ there is a unique order ℓ where xi ≺ℓ xj.

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Generalization

A set of k partial orders ≺1, ..., ≺k partition a tournament D if xi → xj ⇔ there is a unique order ℓ where xi ≺ℓ xj. Every tournament which can be partitioned into k partial orders has a dominating set of size at most f (k). Conjecture (Gy´ arf´ as, P´ alv¨

• lgyi ’14)

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Generalization

A set of k partial orders ≺1, ..., ≺k partition a tournament D if xi → xj ⇔ there is a unique order ℓ where xi ≺ℓ xj. Every tournament which can be partitioned into k partial orders has a dominating set of size at most f (k). Conjecture (Gy´ arf´ as, P´ alv¨

• lgyi ’14)

Bounded VC-dimension ? NO !

complement of random random

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Generalization

A set of k partial orders ≺1, ..., ≺k partition a tournament D if xi → xj ⇔ there is a unique order ℓ where xi ≺ℓ xj. Every tournament which can be partitioned into k partial orders has a dominating set of size at most f (k). Conjecture (Gy´ arf´ as, P´ alv¨

• lgyi ’14)

Bounded VC-dimension ? NO !

complement of random random

Remark : k-majority tournaments can be partitioned into 22k−2 partial orders.

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Related questions

Gy´ arf´ as, P´ alv¨

• lgyi conjecture implies in particular the following :

Every finite subset X of Rd can be covered by f (d) X-boxes (i.e. each box has two antipodal points in X). Theorem (B´ ar´ any and Lehel)

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Related questions

Gy´ arf´ as, P´ alv¨

• lgyi conjecture implies in particular the following :

Every finite subset X of Rd can be covered by f (d) X-boxes (i.e. each box has two antipodal points in X). Theorem (B´ ar´ any and Lehel) GP conjecture is a particular case of : In every k-arc colored tournament, there exist f (k) vertices that can reach any vertex via a monochromatic path. Conjecture (Erd˝

• s, Sands, Sauer, Woodraw ’82)

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Related questions

Gy´ arf´ as, P´ alv¨

• lgyi conjecture implies in particular the following :

Every finite subset X of Rd can be covered by f (d) X-boxes (i.e. each box has two antipodal points in X). Theorem (B´ ar´ any and Lehel) GP conjecture is a particular case of : In every k-arc colored tournament, there exist f (k) vertices that can reach any vertex via a monochromatic path. Conjecture (Erd˝

• s, Sands, Sauer, Woodraw ’82)

B., Lochet, Thomass´ e (2017) : ESSW conjecture is true.

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Sketch of the proof

For any tournament, there exists w : V → R+ such that :

• w(V ) = 1.
• For every v ∈ V , w(N−[v]) ≥ 1/2.

Theorem (Fisher)

G

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Sketch of the proof

For any tournament, there exists w : V → R+ such that :

• w(V ) = 1.
• For every v ∈ V , w(N−[v]) ≥ 1/2.

Theorem (Fisher) ⇒ For any v ∈ V , there exists a color • such that |N−

• [v]| ≥ 1/2k.

G

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Sketch of the proof

For any tournament, there exists w : V → R+ such that :

• w(V ) = 1.
• For every v ∈ V , w(N−[v]) ≥ 1/2.

Theorem (Fisher) ⇒ For any v ∈ V , there exists a color • such that |N−

• [v]| ≥ 1/2k.
• G

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Sketch of the proof

For any tournament, there exists w : V → R+ such that :

• w(V ) = 1.
• For every v ∈ V , w(N−[v]) ≥ 1/2.

Theorem (Fisher) ⇒ For any v ∈ V , there exists a color • such that |N−

• [v]| ≥ 1/2k.
• G

G• G• G• G•

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Sketch of the proof

For any tournament, there exists w : V → R+ such that :

• w(V ) = 1.
• For every v ∈ V , w(N−[v]) ≥ 1/2.

Theorem (Fisher) ⇒ For any v ∈ V , there exists a color • such that |N−

• [v]| ≥ 1/2k.

· · · · · · · · · · · ·

• G

G• G• G• G•

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Sketch of the proof

For any tournament, there exists w : V → R+ such that :

• w(V ) = 1.
• For every v ∈ V , w(N−[v]) ≥ 1/2.

Theorem (Fisher) ⇒ For any v ∈ V , there exists a color • such that |N−

• [v]| ≥ 1/2k.

· · · · · · · · · · · ·

• G

G• G• G• G•

Repeat (k + 1) times : Partition into kk+1 parts.

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Dominating each part

Goal : Show that each part can be dominated using f (k) vertices.

G1, w1 G2, w2 G3

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Dominating each part

Goal : Show that each part can be dominated using f (k) vertices. Idea of the proof :

• After k + 1 extractions, a color • is repeated twice.

G1, w1 G2, w2 G3

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Dominating each part

Goal : Show that each part can be dominated using f (k) vertices. Idea of the proof :

• After k + 1 extractions, a color • is repeated twice.
• x ∈ G3 satisfies w2(N−
• [x]) ≥ 1/2k.

G1, w1 G2, w2 G3 ≥ 1

2k 15/16

Dominating each part

Goal : Show that each part can be dominated using f (k) vertices. Idea of the proof :

• After k + 1 extractions, a color • is repeated twice.
• x ∈ G3 satisfies w2(N−
• [x]) ≥ 1/2k.
• y ∈ G2 satisfies w1(N−
• [y]) ≥ 1/2k.

G1, w1 G2, w2 G3 ≥ 1

2k 15/16

Dominating each part

Goal : Show that each part can be dominated using f (k) vertices. Idea of the proof :

• After k + 1 extractions, a color • is repeated twice.
• x ∈ G3 satisfies w2(N−
• [x]) ≥ 1/2k.
• y ∈ G2 satisfies w1(N−
• [y]) ≥ 1/2k.

G1, w1 G2, w2 G3 ≥ 1

2k

?

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Dominating each part

Goal : Show that each part can be dominated using f (k) vertices. Idea of the proof :

• After k + 1 extractions, a color • is repeated twice.
• x ∈ G3 satisfies w2(N−
• [x]) ≥ 1/2k.
• y ∈ G2 satisfies w1(N−
• [y]) ≥ 1/2k.
• Sample f (k) vertices X in G1 with probability rule w1.

G1, w1 G2, w2 G3 ≥ 1

2k

?

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Dominating each part

Goal : Show that each part can be dominated using f (k) vertices. Idea of the proof :

• After k + 1 extractions, a color • is repeated twice.
• x ∈ G3 satisfies w2(N−
• [x]) ≥ 1/2k.
• y ∈ G2 satisfies w1(N−
• [y]) ≥ 1/2k.
• Sample f (k) vertices X in G1 with probability rule w1.
• Each vertex of G2 has an in-neighbor in X with pr.

≥ (1 − 1/2k).

G1, w1 G2, w2 G3 ≥ 1

2k

?

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Dominating each part

Goal : Show that each part can be dominated using f (k) vertices. Idea of the proof :

• After k + 1 extractions, a color • is repeated twice.
• x ∈ G3 satisfies w2(N−
• [x]) ≥ 1/2k.
• y ∈ G2 satisfies w1(N−
• [y]) ≥ 1/2k.
• Sample f (k) vertices X in G1 with probability rule w1.
• Each vertex of G2 has an in-neighbor in X with pr.

≥ (1 − 1/2k).

• There exists X s.t. w2(N+[X] ∩ G2) ≥ 1 − 1/2k.

G1, w1 G2, w2 G3

f(k) w ≥ (1 − 1/2k)w(G2)

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Dominating each part

Goal : Show that each part can be dominated using f (k) vertices. Idea of the proof :

• After k + 1 extractions, a color • is repeated twice.
• x ∈ G3 satisfies w2(N−
• [x]) ≥ 1/2k.
• y ∈ G2 satisfies w1(N−
• [y]) ≥ 1/2k.
• Sample f (k) vertices X in G1 with probability rule w1.
• Each vertex of G2 has an in-neighbor in X with pr.

≥ (1 − 1/2k).

• There exists X s.t. w2(N+[X] ∩ G2) ≥ 1 − 1/2k.

G1, w1 G2, w2 G3 ≥ 1

2k

f(k) w ≥ (1 − 1/2k)w(G2)

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Conclusion

• There are kk+1 parts.
• Each part can be dominated by O(k · log k) vertices via

monochromatic paths. ⇒ The Erd˝

• s Sands Sauer Woodrow conjecture is satisfied...

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Conclusion

• There are kk+1 parts.
• Each part can be dominated by O(k · log k) vertices via

monochromatic paths. ⇒ The Erd˝

• s Sands Sauer Woodrow conjecture is satisfied... and

monochromatic paths have length at most 2 !

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Conclusion

• There are kk+1 parts.
• Each part can be dominated by O(k · log k) vertices via

monochromatic paths. ⇒ The Erd˝

• s Sands Sauer Woodrow conjecture is satisfied... and

monochromatic paths have length at most 2 ! In every k-arc colored oriented graph, there exist f (k) inde- pendent sets X1, . . . , Xf (k) that “can reach any vertex via a mo- nochromatic path”. Conjecture (Erd˝

• s, Sands, Sauer, Woodraw)

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Conclusion

• There are kk+1 parts.
• Each part can be dominated by O(k · log k) vertices via

monochromatic paths. ⇒ The Erd˝

• s Sands Sauer Woodrow conjecture is satisfied... and

monochromatic paths have length at most 2 ! In every k-arc colored oriented graph, there exist f (k) inde- pendent sets X1, . . . , Xf (k) that “can reach any vertex via a mo- nochromatic path”. Conjecture (Erd˝

• s, Sands, Sauer, Woodraw)

BLT17 : true if the maximum independent set is bounded.

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Conclusion

• There are kk+1 parts.
• Each part can be dominated by O(k · log k) vertices via

monochromatic paths. ⇒ The Erd˝

• s Sands Sauer Woodrow conjecture is satisfied... and

monochromatic paths have length at most 2 ! In every k-arc colored oriented graph, there exist f (k) inde- pendent sets X1, . . . , Xf (k) that “can reach any vertex via a mo- nochromatic path”. Conjecture (Erd˝

• s, Sands, Sauer, Woodraw)

BLT17 : true if the maximum independent set is bounded. Thanks for your attention !

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