Transversals in hypergraphs Anders Yeo AndersYeo@gmail.com - - PowerPoint PPT Presentation

Transversals in hypergraphs

Anders Yeo

AndersYeo@gmail.com Engineering Systems and Design Singapore University of Technology and Design 8 Somapah Road 487372, Singapore

Based on several papers co-authored by people such as M. A. Henning and C. L¨

• wenstein

Anders Yeo Transversals in hypergraphs

This talk

A few definitions and applications. Known results and a number of conjectures. Progress on the Tuza-Vestergaard Conjecture. Progress on a conjecture for linear 4-uniform hypergraphs. Conclusion and further open problems.

Anders Yeo Transversals in hypergraphs

This talk

A few definitions and applications. Known results and a number of conjectures. Progress on the Tuza-Vestergaard Conjecture. Progress on a conjecture for linear 4-uniform hypergraphs. Conclusion and further open problems.

Anders Yeo Transversals in hypergraphs

This talk

A few definitions and applications. Known results and a number of conjectures. Progress on the Tuza-Vestergaard Conjecture. Progress on a conjecture for linear 4-uniform hypergraphs. Conclusion and further open problems.

Anders Yeo Transversals in hypergraphs

This talk

A few definitions and applications. Known results and a number of conjectures. Progress on the Tuza-Vestergaard Conjecture. Progress on a conjecture for linear 4-uniform hypergraphs. Conclusion and further open problems.

Anders Yeo Transversals in hypergraphs

This talk

A few definitions and applications. Known results and a number of conjectures. Progress on the Tuza-Vestergaard Conjecture. Progress on a conjecture for linear 4-uniform hypergraphs. Conclusion and further open problems.

Anders Yeo Transversals in hypergraphs

Definitions

A hypergraph consists of vertices and edges.

x a1 a2 a3 b1 b2 b3

Not linear

x y u v

A linear hypergraph has no pair of vertices overlapping in more than one vertex. A transversal in a hypergraph is a vertex-set intersecting every

• edge. τ(H) denotes the size of a minimum transversal in H.

Anders Yeo Transversals in hypergraphs

Definitions

A hypergraph consists of vertices and edges.

x a1 a2 a3 b1 b2 b3

Not linear

x y u v

A linear hypergraph has no pair of vertices overlapping in more than one vertex. A transversal in a hypergraph is a vertex-set intersecting every

• edge. τ(H) denotes the size of a minimum transversal in H.

Anders Yeo Transversals in hypergraphs

Definitions

A hypergraph consists of vertices and edges.

x a1 a2 a3 b1 b2 b3

Not linear

x y u v

A linear hypergraph has no pair of vertices overlapping in more than one vertex. A transversal in a hypergraph is a vertex-set intersecting every

• edge. τ(H) denotes the size of a minimum transversal in H.

Anders Yeo Transversals in hypergraphs

Definitions

A hypergraph consists of vertices and edges.

x a1 a2 a3 b1 b2 b3

Not linear

x y u v

A linear hypergraph has no pair of vertices overlapping in more than one vertex. A transversal in a hypergraph is a vertex-set intersecting every

• edge. τ(H) denotes the size of a minimum transversal in H.

Anders Yeo Transversals in hypergraphs

Definitions

A hypergraph consists of vertices and edges.

x a1 a2 a3 b1 b2 b3

Not linear

x y u v

A linear hypergraph has no pair of vertices overlapping in more than one vertex. A transversal in a hypergraph is a vertex-set intersecting every

• edge. τ(H) denotes the size of a minimum transversal in H.

Anders Yeo Transversals in hypergraphs

Definitions

A hypergraph consists of vertices and edges.

x a1 a2 a3 b1 b2 b3

Not linear

x y u v

A linear hypergraph has no pair of vertices overlapping in more than one vertex. A transversal in a hypergraph is a vertex-set intersecting every

• edge. τ(H) denotes the size of a minimum transversal in H.

Anders Yeo Transversals in hypergraphs

Definitions

A hypergraph consists of vertices and edges.

x a1 a2 a3 b1 b2 b3

Not linear

x y u v

A linear hypergraph has no pair of vertices overlapping in more than one vertex. A transversal in a hypergraph is a vertex-set intersecting every

• edge. τ(H) denotes the size of a minimum transversal in H.

Anders Yeo Transversals in hypergraphs

Definitions

A hypergraph consists of vertices and edges.

x a1 a2 a3 b1 b2 b3

Not linear

x y u v

A linear hypergraph has no pair of vertices overlapping in more than one vertex. A transversal in a hypergraph is a vertex-set intersecting every

• edge. τ(H) denotes the size of a minimum transversal in H.

Anders Yeo Transversals in hypergraphs

Definitions

A hypergraph consists of vertices and edges.

x a1 a2 a3 b1 b2 b3

Not linear

x y u v

A linear hypergraph has no pair of vertices overlapping in more than one vertex. A transversal in a hypergraph is a vertex-set intersecting every

• edge. τ(H) denotes the size of a minimum transversal in H.

Anders Yeo Transversals in hypergraphs

Definitions

A hypergraph consists of vertices and edges.

x a1 a2 a3 b1 b2 b3

Not linear

x y u v

A linear hypergraph has no pair of vertices overlapping in more than one vertex. A transversal in a hypergraph is a vertex-set intersecting every

• edge. τ(H) denotes the size of a minimum transversal in H.

Anders Yeo Transversals in hypergraphs

Definitions

A hypergraph consists of vertices and edges.

x a1 a2 a3 b1 b2 b3

Not linear

x y u v

A linear hypergraph has no pair of vertices overlapping in more than one vertex. A transversal in a hypergraph is a vertex-set intersecting every

• edge. τ(H) denotes the size of a minimum transversal in H.

Anders Yeo Transversals in hypergraphs

Definitions

A hypergraph consists of vertices and edges.

x a1 a2 a3 b1 b2 b3

Not linear

x y u v

A linear hypergraph has no pair of vertices overlapping in more than one vertex. A transversal in a hypergraph is a vertex-set intersecting every

• edge. τ(H) denotes the size of a minimum transversal in H.

Anders Yeo Transversals in hypergraphs

Definitions

A hypergraph consists of vertices and edges.

x a1 a2 a3 b1 b2 b3

Not linear

x y u v

A linear hypergraph has no pair of vertices overlapping in more than one vertex. A transversal in a hypergraph is a vertex-set intersecting every

• edge. τ(H) denotes the size of a minimum transversal in H.

Anders Yeo Transversals in hypergraphs

Definitions

A hypergraph consists of vertices and edges.

x a1 a2 a3 b1 b2 b3

Not linear

x y u v

A linear hypergraph has no pair of vertices overlapping in more than one vertex. A transversal in a hypergraph is a vertex-set intersecting every

• edge. τ(H) denotes the size of a minimum transversal in H.

Anders Yeo Transversals in hypergraphs

Definitions

A hypergraph consists of vertices and edges.

x a1 a2 a3 b1 b2 b3 a1 a1 a1

Not linear

x y u v x u

A linear hypergraph has no pair of vertices overlapping in more than one vertex. A transversal in a hypergraph is a vertex-set intersecting every

• edge. τ(H) denotes the size of a minimum transversal in H.

Anders Yeo Transversals in hypergraphs

Definitions

x a1 a2 a3 b1 b2 b3 x y u v

The rank of a hypergraph is the size of its largest edge. A hypergraph is k-uniform if all edges have size k. A 2-uniform hypergraph is equivalent to a graph. A hypergraph is r-regular if all verties belong to r edges. That is, they all have degree r.

Anders Yeo Transversals in hypergraphs

Definitions

x a1 a2 a3 b1 b2 b3

rank = 3

x y u v

rank = 3

The rank of a hypergraph is the size of its largest edge. A hypergraph is k-uniform if all edges have size k. A 2-uniform hypergraph is equivalent to a graph. A hypergraph is r-regular if all verties belong to r edges. That is, they all have degree r.

Anders Yeo Transversals in hypergraphs

Definitions

x a1 a2 a3 b1 b2 b3

rank = 3

x y u v

rank = 3

The rank of a hypergraph is the size of its largest edge. A hypergraph is k-uniform if all edges have size k. A 2-uniform hypergraph is equivalent to a graph. A hypergraph is r-regular if all verties belong to r edges. That is, they all have degree r.

Anders Yeo Transversals in hypergraphs

Definitions

x a1 a2 a3 b1 b2 b3

rank = 3 3-uniform

x y u v

rank = 3 Not uniform

The rank of a hypergraph is the size of its largest edge. A hypergraph is k-uniform if all edges have size k. A 2-uniform hypergraph is equivalent to a graph. A hypergraph is r-regular if all verties belong to r edges. That is, they all have degree r.

Anders Yeo Transversals in hypergraphs

Definitions

x a1 a2 a3 b1 b2 b3

rank = 3 3-uniform

x y u v

rank = 3 Not uniform

The rank of a hypergraph is the size of its largest edge. A hypergraph is k-uniform if all edges have size k. A 2-uniform hypergraph is equivalent to a graph. A hypergraph is r-regular if all verties belong to r edges. That is, they all have degree r.

Anders Yeo Transversals in hypergraphs

Definitions

x a1 a2 a3 b1 b2 b3

rank = 3 3-uniform Not regular

x y u v

rank = 3 Not uniform 2-regular

The rank of a hypergraph is the size of its largest edge. A hypergraph is k-uniform if all edges have size k. A 2-uniform hypergraph is equivalent to a graph. A hypergraph is r-regular if all verties belong to r edges. That is, they all have degree r.

Anders Yeo Transversals in hypergraphs

Applications

3-SAT Colinear points in the plane. Total domination Phylogenetic trees in Biology. Feedback vertex sets (in tournaments). Coloring in hypergraphs. 2-coloring is equivalent to vertex-disjoint transversals. And many more..... Why?

Anders Yeo Transversals in hypergraphs

Applications

3-SAT Colinear points in the plane. Total domination Phylogenetic trees in Biology. Feedback vertex sets (in tournaments). Coloring in hypergraphs. 2-coloring is equivalent to vertex-disjoint transversals. And many more..... Why?

Anders Yeo Transversals in hypergraphs

Applications

3-SAT Colinear points in the plane. Total domination Phylogenetic trees in Biology. Feedback vertex sets (in tournaments). Coloring in hypergraphs. 2-coloring is equivalent to vertex-disjoint transversals. And many more..... Why?

Anders Yeo Transversals in hypergraphs

Applications

3-SAT Colinear points in the plane. Total domination Phylogenetic trees in Biology. Feedback vertex sets (in tournaments). Coloring in hypergraphs. 2-coloring is equivalent to vertex-disjoint transversals. And many more..... Why?

Anders Yeo Transversals in hypergraphs

Applications

3-SAT Colinear points in the plane. Total domination Phylogenetic trees in Biology. Feedback vertex sets (in tournaments). Coloring in hypergraphs. 2-coloring is equivalent to vertex-disjoint transversals. And many more..... Why?

Anders Yeo Transversals in hypergraphs

Applications

3-SAT Colinear points in the plane. Total domination Phylogenetic trees in Biology. Feedback vertex sets (in tournaments). Coloring in hypergraphs. 2-coloring is equivalent to vertex-disjoint transversals. And many more..... Why?

Anders Yeo Transversals in hypergraphs

Applications

3-SAT Colinear points in the plane. Total domination Phylogenetic trees in Biology. Feedback vertex sets (in tournaments). Coloring in hypergraphs. 2-coloring is equivalent to vertex-disjoint transversals. And many more..... Why?

Anders Yeo Transversals in hypergraphs

Applications

3-SAT Colinear points in the plane. Total domination Phylogenetic trees in Biology. Feedback vertex sets (in tournaments). Coloring in hypergraphs. 2-coloring is equivalent to vertex-disjoint transversals. And many more..... Why?

Anders Yeo Transversals in hypergraphs

3-SAT

Consider an instance, I, of 3-SAT with n vriables and m clauses. For example, I = (v1 ∨ v2 ∨ v3) ∧ (v2 ∨ v3 ∨ v4) ∧ · · · . · · ·

v1 v1 v2 v2 v3 v3 v4 v4 v5 v5 vn vn

We build a hypergraph, H, with 2n vertices as above. Each variable gives rise to a 2-edge. Each clause gives rise to a 3-edge. H has a transversal of size n if and only if I is satisfiable. We believe this reduction may be used to improve currently best algorithms for 3-SAT.

Anders Yeo Transversals in hypergraphs

3-SAT

Consider an instance, I, of 3-SAT with n vriables and m clauses. For example, I = (v1 ∨ v2 ∨ v3) ∧ (v2 ∨ v3 ∨ v4) ∧ · · · . · · ·

v1 v1 v2 v2 v3 v3 v4 v4 v5 v5 vn vn

We build a hypergraph, H, with 2n vertices as above. Each variable gives rise to a 2-edge. Each clause gives rise to a 3-edge. H has a transversal of size n if and only if I is satisfiable. We believe this reduction may be used to improve currently best algorithms for 3-SAT.

Anders Yeo Transversals in hypergraphs

3-SAT

Consider an instance, I, of 3-SAT with n vriables and m clauses. For example, I = (v1 ∨ v2 ∨ v3) ∧ (v2 ∨ v3 ∨ v4) ∧ · · · . · · ·

v1 v1 v2 v2 v3 v3 v4 v4 v5 v5 vn vn

We build a hypergraph, H, with 2n vertices as above. Each variable gives rise to a 2-edge. Each clause gives rise to a 3-edge. H has a transversal of size n if and only if I is satisfiable. We believe this reduction may be used to improve currently best algorithms for 3-SAT.

Anders Yeo Transversals in hypergraphs

3-SAT

Consider an instance, I, of 3-SAT with n vriables and m clauses. For example, I = (v1 ∨ v2 ∨ v3) ∧ (v2 ∨ v3 ∨ v4) ∧ · · · . · · ·

v1 v1 v2 v2 v3 v3 v4 v4 v5 v5 vn vn

We build a hypergraph, H, with 2n vertices as above. Each variable gives rise to a 2-edge. Each clause gives rise to a 3-edge. H has a transversal of size n if and only if I is satisfiable. We believe this reduction may be used to improve currently best algorithms for 3-SAT.

Anders Yeo Transversals in hypergraphs

3-SAT

Consider an instance, I, of 3-SAT with n vriables and m clauses. For example, I = (v1 ∨ v2 ∨ v3) ∧ (v2 ∨ v3 ∨ v4) ∧ · · · . · · ·

v1 v1 v2 v2 v3 v3 v4 v4 v5 v5 vn vn

We build a hypergraph, H, with 2n vertices as above. Each variable gives rise to a 2-edge. Each clause gives rise to a 3-edge. H has a transversal of size n if and only if I is satisfiable. We believe this reduction may be used to improve currently best algorithms for 3-SAT.

Anders Yeo Transversals in hypergraphs

3-SAT

Consider an instance, I, of 3-SAT with n vriables and m clauses. For example, I = (v1 ∨ v2 ∨ v3) ∧ (v2 ∨ v3 ∨ v4) ∧ · · · . · · ·

v1 v1 v2 v2 v3 v3 v4 v4 v5 v5 vn vn

We build a hypergraph, H, with 2n vertices as above. Each variable gives rise to a 2-edge. Each clause gives rise to a 3-edge. H has a transversal of size n if and only if I is satisfiable. We believe this reduction may be used to improve currently best algorithms for 3-SAT.

Anders Yeo Transversals in hypergraphs

3-SAT

Consider an instance, I, of 3-SAT with n vriables and m clauses. For example, I = (v1 ∨ v2 ∨ v3) ∧ (v2 ∨ v3 ∨ v4) ∧ · · · . · · ·

v1 v1 v2 v2 v3 v3 v4 v4 v5 v5 vn vn

We build a hypergraph, H, with 2n vertices as above. Each variable gives rise to a 2-edge. Each clause gives rise to a 3-edge. H has a transversal of size n if and only if I is satisfiable. We believe this reduction may be used to improve currently best algorithms for 3-SAT.

Anders Yeo Transversals in hypergraphs

Colinear points in the plane

We are given a number of points in the plane. Can we find the largest set of points where no 3 are co-linear (lie

• n a line)?

Add 3-edges between all sets of 3 points that are colinear. Remove the points of a minimum transversal. If no four points are co-linear then the resulting hypergraph is linear.

Anders Yeo Transversals in hypergraphs

Colinear points in the plane

We are given a number of points in the plane. Can we find the largest set of points where no 3 are co-linear (lie

• n a line)?

Add 3-edges between all sets of 3 points that are colinear. Remove the points of a minimum transversal. If no four points are co-linear then the resulting hypergraph is linear.

Anders Yeo Transversals in hypergraphs

Colinear points in the plane

We are given a number of points in the plane. Can we find the largest set of points where no 3 are co-linear (lie

• n a line)?

Add 3-edges between all sets of 3 points that are colinear. Remove the points of a minimum transversal. If no four points are co-linear then the resulting hypergraph is linear.

Anders Yeo Transversals in hypergraphs

Colinear points in the plane

We are given a number of points in the plane. Can we find the largest set of points where no 3 are co-linear (lie

• n a line)?

Add 3-edges between all sets of 3 points that are colinear. Remove the points of a minimum transversal. If no four points are co-linear then the resulting hypergraph is linear.

Anders Yeo Transversals in hypergraphs

Colinear points in the plane

We are given a number of points in the plane. Can we find the largest set of points where no 3 are co-linear (lie

• n a line)?

Add 3-edges between all sets of 3 points that are colinear. Remove the points of a minimum transversal. If no four points are co-linear then the resulting hypergraph is linear.

Anders Yeo Transversals in hypergraphs

Colinear points in the plane

We are given a number of points in the plane. Can we find the largest set of points where no 3 are co-linear (lie

• n a line)?

Add 3-edges between all sets of 3 points that are colinear. Remove the points of a minimum transversal. If no four points are co-linear then the resulting hypergraph is linear.

Anders Yeo Transversals in hypergraphs

Colinear points in the plane

We are given a number of points in the plane. Can we find the largest set of points where no 3 are co-linear (lie

• n a line)?

Add 3-edges between all sets of 3 points that are colinear. Remove the points of a minimum transversal. If no four points are co-linear then the resulting hypergraph is linear.

Anders Yeo Transversals in hypergraphs

Total Domination

S is a total dominating set in G, if and only if every vertex in G has a neighbour in S. G

u1 u2 u3 u4 u5

G ′

v1 v2 v3 v4 v5 v6

γt(G) denotes the size of a smallest total dominating set in G.

Anders Yeo Transversals in hypergraphs

Total Domination

S is a total dominating set in G, if and only if every vertex in G has a neighbour in S. G

u1 u2 u3 u4 u5

G ′

v1 v2 v3 v4 v5 v6

γt(G) denotes the size of a smallest total dominating set in G.

Anders Yeo Transversals in hypergraphs

Total Domination

S is a total dominating set in G, if and only if every vertex in G has a neighbour in S. G

u1 u2 u3 u4 u5

G ′

v1 v2 v3 v4 v5 v6

γt(G) denotes the size of a smallest total dominating set in G.

Anders Yeo Transversals in hypergraphs

Total Domination

S is a total dominating set in G, if and only if every vertex in G has a neighbour in S. G

u1 u2 u3 u4 u5

G ′

v1 v2 v3 v4 v5 v6

γt(G) denotes the size of a smallest total dominating set in G.

Anders Yeo Transversals in hypergraphs

Total Domination

S is a total dominating set in G, if and only if every vertex in G has a neighbour in S. G

u1 u2 u3 u4 u5

G ′

v1 v2 v3 v4 v5 v6

γt(G) denotes the size of a smallest total dominating set in G.

Anders Yeo Transversals in hypergraphs

Total Domination

S is a total dominating set in G, if and only if every vertex in G has a neighbour in S. G

u1 u2 u3 u4 u5

G ′

v1 v2 v3 v4 v5 v6

γt(G) denotes the size of a smallest total dominating set in G.

Anders Yeo Transversals in hypergraphs

Total Domination

S is a total dominating set in G, if and only if every vertex in G has a neighbour in S. G

u1 u2 u3 u4 u5

G ′

v1 v2 v3 v4 v5 v6

γt(G) denotes the size of a smallest total dominating set in G.

Anders Yeo Transversals in hypergraphs

Total Domination

S is a total dominating set in G, if and only if every vertex in G has a neighbour in S. G

u1 u2 u3 u4 u5

G ′

v1 v2 v3 v4 v5 v6

γt(G) denotes the size of a smallest total dominating set in G.

Anders Yeo Transversals in hypergraphs

Total Domination, Open Neighbourhood hypergraph

Given G, the Open Neighbourhood Hypergraph, ONH(G), of G has vertex-set V (G) and edge-set {N(v) | v ∈ V (G)}. G

u1 u2 u3 u4 u5

ONH(G)

u1 u2 u3 u4 u5

A transversal in ONH(G) is equivalent to a total dominating set in G. Most recent results on Total Domination are found using transversals in hypergraphs.

Anders Yeo Transversals in hypergraphs

Total Domination, Open Neighbourhood hypergraph

Given G, the Open Neighbourhood Hypergraph, ONH(G), of G has vertex-set V (G) and edge-set {N(v) | v ∈ V (G)}. G

u1 u2 u3 u4 u5

ONH(G)

u1 u2 u3 u4 u5

A transversal in ONH(G) is equivalent to a total dominating set in G. Most recent results on Total Domination are found using transversals in hypergraphs.

Anders Yeo Transversals in hypergraphs

Total Domination, Open Neighbourhood hypergraph

Given G, the Open Neighbourhood Hypergraph, ONH(G), of G has vertex-set V (G) and edge-set {N(v) | v ∈ V (G)}. G

u1 u2 u3 u4 u5

ONH(G)

u1 u2 u3 u4 u5

A transversal in ONH(G) is equivalent to a total dominating set in G. Most recent results on Total Domination are found using transversals in hypergraphs.

Anders Yeo Transversals in hypergraphs

Total Domination, Open Neighbourhood hypergraph

Given G, the Open Neighbourhood Hypergraph, ONH(G), of G has vertex-set V (G) and edge-set {N(v) | v ∈ V (G)}. G

u1 u2 u3 u4 u5

ONH(G)

u1 u2 u3 u4 u5

A transversal in ONH(G) is equivalent to a total dominating set in G. Most recent results on Total Domination are found using transversals in hypergraphs.

Anders Yeo Transversals in hypergraphs

Total Domination, Open Neighbourhood hypergraph

Given G, the Open Neighbourhood Hypergraph, ONH(G), of G has vertex-set V (G) and edge-set {N(v) | v ∈ V (G)}. G

u1 u2 u3 u4 u5

ONH(G)

u1 u2 u3 u4 u5

A transversal in ONH(G) is equivalent to a total dominating set in G. Most recent results on Total Domination are found using transversals in hypergraphs.

Anders Yeo Transversals in hypergraphs

Total Domination, Open Neighbourhood hypergraph

Given G, the Open Neighbourhood Hypergraph, ONH(G), of G has vertex-set V (G) and edge-set {N(v) | v ∈ V (G)}. G

u1 u2 u3 u4 u5

ONH(G)

u1 u2 u3 u4 u5

A transversal in ONH(G) is equivalent to a total dominating set in G. Most recent results on Total Domination are found using transversals in hypergraphs.

Anders Yeo Transversals in hypergraphs

Total Domination, Open Neighbourhood hypergraph

Given G, the Open Neighbourhood Hypergraph, ONH(G), of G has vertex-set V (G) and edge-set {N(v) | v ∈ V (G)}. G

u1 u2 u3 u4 u5

ONH(G)

u1 u2 u3 u4 u5

A transversal in ONH(G) is equivalent to a total dominating set in G. Most recent results on Total Domination are found using transversals in hypergraphs.

Anders Yeo Transversals in hypergraphs

Total Domination, Open Neighbourhood hypergraph

Given G, the Open Neighbourhood Hypergraph, ONH(G), of G has vertex-set V (G) and edge-set {N(v) | v ∈ V (G)}. G

u1 u2 u3 u4 u5

ONH(G)

u1 u2 u3 u4 u5

A transversal in ONH(G) is equivalent to a total dominating set in G. Most recent results on Total Domination are found using transversals in hypergraphs.

Anders Yeo Transversals in hypergraphs

More....

In Phylogenetic trees we want to remove as few species as possible such that the known Phylogenetic trees dont contradict each other. This can be modelled as a transversal problem in 3-uniform hypergraphs. Feedback vertex sets (in tournaments). Given a graph or digraph put a hyperdege around the vertex set of all chordless

• cycles. A transversal in the resulting hypergraph is a minimum

feedback vertex set. For tournaments all chordless cycles are 3-cycles so our hypergraph is 3-uniform. We have best known algorithms for both the above using transversals in hypergraphs.

Anders Yeo Transversals in hypergraphs

More....

In Phylogenetic trees we want to remove as few species as possible such that the known Phylogenetic trees dont contradict each other. This can be modelled as a transversal problem in 3-uniform hypergraphs. Feedback vertex sets (in tournaments). Given a graph or digraph put a hyperdege around the vertex set of all chordless

• cycles. A transversal in the resulting hypergraph is a minimum

feedback vertex set. For tournaments all chordless cycles are 3-cycles so our hypergraph is 3-uniform. We have best known algorithms for both the above using transversals in hypergraphs.

Anders Yeo Transversals in hypergraphs

More....

In Phylogenetic trees we want to remove as few species as possible such that the known Phylogenetic trees dont contradict each other. This can be modelled as a transversal problem in 3-uniform hypergraphs. Feedback vertex sets (in tournaments). Given a graph or digraph put a hyperdege around the vertex set of all chordless

• cycles. A transversal in the resulting hypergraph is a minimum

feedback vertex set. For tournaments all chordless cycles are 3-cycles so our hypergraph is 3-uniform. We have best known algorithms for both the above using transversals in hypergraphs.

Anders Yeo Transversals in hypergraphs

Known results and conjectures, order and size

For connected k-uniform hypergraphs there are bounds just based

• n the order, n, and size, m.

(Erd¨

• s & Tuza) If k = 2 then τ(H) ≤ 2(n+m+1)

7

. (Chv´ atal & McDiarmid) If k ≥ 2 then τ(H) ≤ n+⌊k/2⌋m

⌊3k/2⌋ .

The above bounds are tight for hy- pergraphs of average degree 2. For larger average degrees not much is known.

k 2 k 2 k 2

n = 3k/2 m = 3 τ = 2

The following is tight for average degrees 3-4. (Thomass´ e & Yeo) If k = 4 then τ(H) ≤ 5n+4m

21

.

Anders Yeo Transversals in hypergraphs

Known results and conjectures, order and size

For connected k-uniform hypergraphs there are bounds just based

• n the order, n, and size, m.

(Erd¨

• s & Tuza) If k = 2 then τ(H) ≤ 2(n+m+1)

7

. (Chv´ atal & McDiarmid) If k ≥ 2 then τ(H) ≤ n+⌊k/2⌋m

⌊3k/2⌋ .

The above bounds are tight for hy- pergraphs of average degree 2. For larger average degrees not much is known.

k 2 k 2 k 2

n = 3k/2 m = 3 τ = 2

The following is tight for average degrees 3-4. (Thomass´ e & Yeo) If k = 4 then τ(H) ≤ 5n+4m

21

.

Anders Yeo Transversals in hypergraphs

Known results and conjectures, order and size

For connected k-uniform hypergraphs there are bounds just based

• n the order, n, and size, m.

(Erd¨

• s & Tuza) If k = 2 then τ(H) ≤ 2(n+m+1)

7

. (Chv´ atal & McDiarmid) If k ≥ 2 then τ(H) ≤ n+⌊k/2⌋m

⌊3k/2⌋ .

The above bounds are tight for hy- pergraphs of average degree 2. For larger average degrees not much is known.

k 2 k 2 k 2

n = 3k/2 m = 3 τ = 2

The following is tight for average degrees 3-4. (Thomass´ e & Yeo) If k = 4 then τ(H) ≤ 5n+4m

21

.

Anders Yeo Transversals in hypergraphs

Known results and conjectures, order and size

For connected k-uniform hypergraphs there are bounds just based

• n the order, n, and size, m.

(Erd¨

• s & Tuza) If k = 2 then τ(H) ≤ 2(n+m+1)

7

. (Chv´ atal & McDiarmid) If k ≥ 2 then τ(H) ≤ n+⌊k/2⌋m

⌊3k/2⌋ .

The above bounds are tight for hy- pergraphs of average degree 2. For larger average degrees not much is known.

k 2 k 2 k 2

n = 3k/2 m = 3 τ = 2

The following is tight for average degrees 3-4. (Thomass´ e & Yeo) If k = 4 then τ(H) ≤ 5n+4m

21

.

Anders Yeo Transversals in hypergraphs

Known results and conjectures, corollaries

Given the results on the previous slide, we automatically get results for total domination in graphs with given minimum degree. If k = 3 and n = m then τ(H) ≤ n/2. So, γt(G) ≤ n/4 when δ(G) ≥ 3. If k = 4 and n = m then τ(H) ≤ 3n/7. So, γt(G) ≤ 3n/7 when δ(G) ≥ 4. Conjecture: If k = 5 and n = m then τ(H) ≤ 4n/11. The above conjecture would be best possible. Open Problem: For k ≥ 4 prove better bounds for linear hypergraphs.

Anders Yeo Transversals in hypergraphs

Known results and conjectures, corollaries

Given the results on the previous slide, we automatically get results for total domination in graphs with given minimum degree. If k = 3 and n = m then τ(H) ≤ n/2. So, γt(G) ≤ n/4 when δ(G) ≥ 3. If k = 4 and n = m then τ(H) ≤ 3n/7. So, γt(G) ≤ 3n/7 when δ(G) ≥ 4. Conjecture: If k = 5 and n = m then τ(H) ≤ 4n/11. The above conjecture would be best possible. Open Problem: For k ≥ 4 prove better bounds for linear hypergraphs.

Anders Yeo Transversals in hypergraphs

Known results and conjectures, corollaries

Given the results on the previous slide, we automatically get results for total domination in graphs with given minimum degree. If k = 3 and n = m then τ(H) ≤ n/2. So, γt(G) ≤ n/4 when δ(G) ≥ 3. If k = 4 and n = m then τ(H) ≤ 3n/7. So, γt(G) ≤ 3n/7 when δ(G) ≥ 4. Conjecture: If k = 5 and n = m then τ(H) ≤ 4n/11. The above conjecture would be best possible. Open Problem: For k ≥ 4 prove better bounds for linear hypergraphs.

Anders Yeo Transversals in hypergraphs

Known results and conjectures, corollaries

Given the results on the previous slide, we automatically get results for total domination in graphs with given minimum degree. If k = 3 and n = m then τ(H) ≤ n/2. So, γt(G) ≤ n/4 when δ(G) ≥ 3. If k = 4 and n = m then τ(H) ≤ 3n/7. So, γt(G) ≤ 3n/7 when δ(G) ≥ 4. Conjecture: If k = 5 and n = m then τ(H) ≤ 4n/11. The above conjecture would be best possible. Open Problem: For k ≥ 4 prove better bounds for linear hypergraphs.

Anders Yeo Transversals in hypergraphs

Known results and conjectures, corollaries

Given the results on the previous slide, we automatically get results for total domination in graphs with given minimum degree. If k = 3 and n = m then τ(H) ≤ n/2. So, γt(G) ≤ n/4 when δ(G) ≥ 3. If k = 4 and n = m then τ(H) ≤ 3n/7. So, γt(G) ≤ 3n/7 when δ(G) ≥ 4. Conjecture: If k = 5 and n = m then τ(H) ≤ 4n/11. The above conjecture would be best possible. Open Problem: For k ≥ 4 prove better bounds for linear hypergraphs.

Anders Yeo Transversals in hypergraphs

Known results and conjectures, corollaries

Given the results on the previous slide, we automatically get results for total domination in graphs with given minimum degree. If k = 3 and n = m then τ(H) ≤ n/2. So, γt(G) ≤ n/4 when δ(G) ≥ 3. If k = 4 and n = m then τ(H) ≤ 3n/7. So, γt(G) ≤ 3n/7 when δ(G) ≥ 4. Conjecture: If k = 5 and n = m then τ(H) ≤ 4n/11. The above conjecture would be best possible. Open Problem: For k ≥ 4 prove better bounds for linear hypergraphs.

Anders Yeo Transversals in hypergraphs

Known results and conjectures, regularity

Very little is known regarding the regularity of a hypergraphs and the following has been conjectured. Conjecture For all k ≥ 2, if H is a k-uniform hypergraph, then the following holds. (a) If H is 3-regular then τ(H) ≤ 3n

2k (holds for k ≤ 4).

(b) If H is 4-regular then τ(H) ≤ 12n

7k (holds for k ≤ 4).

(c) If H is 5-regular then τ(H) ≤ 20n

11k (holds for k ≤ 3).

(d) If H is 6-regular then τ(H) ≤ 24n

13k (holds for k ≤ 2).

We now believe we have a proof of (a) for when k = 6. That is we can prove, Tuza-Vestergaard Conjecture (Now proved?): Every 3-regular 6-uniform hypergraph, H, satisfies τ(H) ≤ |V (H)|/4.

Anders Yeo Transversals in hypergraphs

Known results and conjectures, regularity

Very little is known regarding the regularity of a hypergraphs and the following has been conjectured. Conjecture For all k ≥ 2, if H is a k-uniform hypergraph, then the following holds. (a) If H is 3-regular then τ(H) ≤ 3n

2k (holds for k ≤ 4).

(b) If H is 4-regular then τ(H) ≤ 12n

7k (holds for k ≤ 4).

(c) If H is 5-regular then τ(H) ≤ 20n

11k (holds for k ≤ 3).

(d) If H is 6-regular then τ(H) ≤ 24n

13k (holds for k ≤ 2).

We now believe we have a proof of (a) for when k = 6. That is we can prove, Tuza-Vestergaard Conjecture (Now proved?): Every 3-regular 6-uniform hypergraph, H, satisfies τ(H) ≤ |V (H)|/4.

Anders Yeo Transversals in hypergraphs

Known results and conjectures, regularity

Very little is known regarding the regularity of a hypergraphs and the following has been conjectured. Conjecture For all k ≥ 2, if H is a k-uniform hypergraph, then the following holds. (a) If H is 3-regular then τ(H) ≤ 3n

2k (holds for k ≤ 4).

(b) If H is 4-regular then τ(H) ≤ 12n

7k (holds for k ≤ 4).

(c) If H is 5-regular then τ(H) ≤ 20n

11k (holds for k ≤ 3).

(d) If H is 6-regular then τ(H) ≤ 24n

13k (holds for k ≤ 2).

We now believe we have a proof of (a) for when k = 6. That is we can prove, Tuza-Vestergaard Conjecture (Now proved?): Every 3-regular 6-uniform hypergraph, H, satisfies τ(H) ≤ |V (H)|/4.

Anders Yeo Transversals in hypergraphs

Known results and conjectures, regularity

Very little is known regarding the regularity of a hypergraphs and the following has been conjectured. Conjecture For all k ≥ 2, if H is a k-uniform hypergraph, then the following holds. (a) If H is 3-regular then τ(H) ≤ 3n

2k (holds for k ≤ 4).

(b) If H is 4-regular then τ(H) ≤ 12n

7k (holds for k ≤ 4).

(c) If H is 5-regular then τ(H) ≤ 20n

11k (holds for k ≤ 3).

(d) If H is 6-regular then τ(H) ≤ 24n

13k (holds for k ≤ 2).

We now believe we have a proof of (a) for when k = 6. That is we can prove, Tuza-Vestergaard Conjecture (Now proved?): Every 3-regular 6-uniform hypergraph, H, satisfies τ(H) ≤ |V (H)|/4.

Anders Yeo Transversals in hypergraphs

Progress on the Tuza-Vestergaard Conjecture

The well-known Tuza-Vestergaard conjecture was published in 2002. Bounds of 0.27778n, 0.27037n and 0.268116n have been showed. There are infinitely many hypergraphs achieving equality in the bound n/4. (Take any 3-regular 3-uniform hyper- graph, H, with τ(H) = n/2 and du- plicate every vertex). Our proof of the Tuza-Vestergaard conjecture also implies a number of other results. For example, the following for k = 6. Conjecture: For all k ≥ 2, if H is a k-uniform hypergraph with maximum degree ∆(H) ≤ 3 then τ(H) ≤ |V (H)|

k

+ |E(H)|

6

.

Anders Yeo Transversals in hypergraphs

Progress on the Tuza-Vestergaard Conjecture

The well-known Tuza-Vestergaard conjecture was published in 2002. Bounds of 0.27778n, 0.27037n and 0.268116n have been showed. There are infinitely many hypergraphs achieving equality in the bound n/4. (Take any 3-regular 3-uniform hyper- graph, H, with τ(H) = n/2 and du- plicate every vertex). Our proof of the Tuza-Vestergaard conjecture also implies a number of other results. For example, the following for k = 6. Conjecture: For all k ≥ 2, if H is a k-uniform hypergraph with maximum degree ∆(H) ≤ 3 then τ(H) ≤ |V (H)|

k

+ |E(H)|

6

.

Anders Yeo Transversals in hypergraphs

Progress on the Tuza-Vestergaard Conjecture

The well-known Tuza-Vestergaard conjecture was published in 2002. Bounds of 0.27778n, 0.27037n and 0.268116n have been showed. There are infinitely many hypergraphs achieving equality in the bound n/4. (Take any 3-regular 3-uniform hyper- graph, H, with τ(H) = n/2 and du- plicate every vertex). Our proof of the Tuza-Vestergaard conjecture also implies a number of other results. For example, the following for k = 6. Conjecture: For all k ≥ 2, if H is a k-uniform hypergraph with maximum degree ∆(H) ≤ 3 then τ(H) ≤ |V (H)|

k

+ |E(H)|

6

.

Anders Yeo Transversals in hypergraphs

Progress on the Tuza-Vestergaard Conjecture

The well-known Tuza-Vestergaard conjecture was published in 2002. Bounds of 0.27778n, 0.27037n and 0.268116n have been showed. There are infinitely many hypergraphs achieving equality in the bound n/4. (Take any 3-regular 3-uniform hyper- graph, H, with τ(H) = n/2 and du- plicate every vertex). Our proof of the Tuza-Vestergaard conjecture also implies a number of other results. For example, the following for k = 6. Conjecture: For all k ≥ 2, if H is a k-uniform hypergraph with maximum degree ∆(H) ≤ 3 then τ(H) ≤ |V (H)|

k

+ |E(H)|

6

.

Anders Yeo Transversals in hypergraphs

Progress on the Tuza-Vestergaard Conjecture

The well-known Tuza-Vestergaard conjecture was published in 2002. Bounds of 0.27778n, 0.27037n and 0.268116n have been showed. There are infinitely many hypergraphs achieving equality in the bound n/4. (Take any 3-regular 3-uniform hyper- graph, H, with τ(H) = n/2 and du- plicate every vertex). Our proof of the Tuza-Vestergaard conjecture also implies a number of other results. For example, the following for k = 6. Conjecture: For all k ≥ 2, if H is a k-uniform hypergraph with maximum degree ∆(H) ≤ 3 then τ(H) ≤ |V (H)|

k

+ |E(H)|

6

.

Anders Yeo Transversals in hypergraphs

Progress on the Tuza-Vestergaard Conjecture

In order to prove the Tuza-Vestergaard Conjecture we prove a much stronger result. If H is a 6-uniform hypergraph on n vertices and m edges with ∆(H) ≤ 3, then 20τ(H) ≤ 2n + 6m + ρ, where ρ is a number that depends on the existence of certain hypergraphs, which will be described in more detail below. SB1 B1 B2 B3 B4

Anders Yeo Transversals in hypergraphs

Progress on the Tuza-Vestergaard Conjecture

In order to prove the Tuza-Vestergaard Conjecture we prove a much stronger result. If H is a 6-uniform hypergraph on n vertices and m edges with ∆(H) ≤ 3, then 20τ(H) ≤ 2n + 6m + ρ, where ρ is a number that depends on the existence of certain hypergraphs, which will be described in more detail below. SB1 B1 B2 B3 B4

Anders Yeo Transversals in hypergraphs

Progress on the Tuza-Vestergaard Conjecture

We recusively define an infinite family B of bad hypergraphs as follows. Let B = {B1, B2, B3, B4} and recursively do the following....

In B →

No minimum transversal contains a red and green vertex. Any hypergraph in B is 2 short in the formula 20τ ≤ 2n + 6m. What about SB1?

Anders Yeo Transversals in hypergraphs

Progress on the Tuza-Vestergaard Conjecture

We recusively define an infinite family B of bad hypergraphs as follows. Let B = {B1, B2, B3, B4} and recursively do the following....

In B →

No minimum transversal contains a red and green vertex. Any hypergraph in B is 2 short in the formula 20τ ≤ 2n + 6m. What about SB1?

Anders Yeo Transversals in hypergraphs

Progress on the Tuza-Vestergaard Conjecture

We recusively define an infinite family B of bad hypergraphs as follows. Let B = {B1, B2, B3, B4} and recursively do the following....

In B →

No minimum transversal contains a red and green vertex. Any hypergraph in B is 2 short in the formula 20τ ≤ 2n + 6m. What about SB1?

Anders Yeo Transversals in hypergraphs

Progress on the Tuza-Vestergaard Conjecture

We recusively define an infinite family B of bad hypergraphs as follows. Let B = {B1, B2, B3, B4} and recursively do the following....

In B →

No minimum transversal contains a red and green vertex. Any hypergraph in B is 2 short in the formula 20τ ≤ 2n + 6m. What about SB1?

Anders Yeo Transversals in hypergraphs

Progress on the Tuza-Vestergaard Conjecture

We recusively define an infinite family B of bad hypergraphs as follows. Let B = {B1, B2, B3, B4} and recursively do the following....

In B →

No minimum transversal contains a red and green vertex. Any hypergraph in B is 2 short in the formula 20τ ≤ 2n + 6m. What about SB1?

Anders Yeo Transversals in hypergraphs

Progress on the Tuza-Vestergaard Conjecture

We recusively define an infinite family B of bad hypergraphs as follows. Let B = {B1, B2, B3, B4} and recursively do the following....

In B →

No minimum transversal contains a red and green vertex. Any hypergraph in B is 2 short in the formula 20τ ≤ 2n + 6m. What about SB1?

Anders Yeo Transversals in hypergraphs

Progress on the Tuza-Vestergaard Conjecture

We recusively define an infinite family B of bad hypergraphs as follows. Let B = {B1, B2, B3, B4} and recursively do the following....

In B →

No minimum transversal contains a red and green vertex. Any hypergraph in B is 2 short in the formula 20τ ≤ 2n + 6m. What about SB1?

Anders Yeo Transversals in hypergraphs

Progress on the Tuza-Vestergaard Conjecture

We recusively define an infinite family B of bad hypergraphs as follows. Let B = {B1, B2, B3, B4} and recursively do the following....

In B →

No minimum transversal contains a red and green vertex. Any hypergraph in B is 2 short in the formula 20τ ≤ 2n + 6m. What about SB1?

Anders Yeo Transversals in hypergraphs

Progress on the Tuza-Vestergaard Conjecture

SB1 is 4 short in the formula 20τ ≤ 2n + 6m. If C is a SB1 subgraph in H, then define f (C) as follows: # degree 2’s in C 1 2 3 4 5 6 7 8 9 f (C) 1 1 1 2 2 3 4 If R ∈ B then f (R) =    2 R is a component in H 1 R is intersected by 1 edge R is intersected by ≥ 2 edges . Let ρ = max

Q∈Q f (Q) over all families of vertex-disjoint

subgraphs Q in H (F(Q) = 0 if Q ∈ B ∪ SB1).

Anders Yeo Transversals in hypergraphs

Progress on the Tuza-Vestergaard Conjecture

SB1 is 4 short in the formula 20τ ≤ 2n + 6m. If C is a SB1 subgraph in H, then define f (C) as follows: # degree 2’s in C 1 2 3 4 5 6 7 8 9 f (C) 1 1 1 2 2 3 4 If R ∈ B then f (R) =    2 R is a component in H 1 R is intersected by 1 edge R is intersected by ≥ 2 edges . Let ρ = max

Q∈Q f (Q) over all families of vertex-disjoint

subgraphs Q in H (F(Q) = 0 if Q ∈ B ∪ SB1).

Anders Yeo Transversals in hypergraphs

Progress on the Tuza-Vestergaard Conjecture

SB1 is 4 short in the formula 20τ ≤ 2n + 6m. If C is a SB1 subgraph in H, then define f (C) as follows: # degree 2’s in C 1 2 3 4 5 6 7 8 9 f (C) 1 1 1 2 2 3 4 If R ∈ B then f (R) =    2 R is a component in H 1 R is intersected by 1 edge R is intersected by ≥ 2 edges . Let ρ = max

Q∈Q f (Q) over all families of vertex-disjoint

subgraphs Q in H (F(Q) = 0 if Q ∈ B ∪ SB1).

Anders Yeo Transversals in hypergraphs

Progress on the Tuza-Vestergaard Conjecture

SB1 is 4 short in the formula 20τ ≤ 2n + 6m. If C is a SB1 subgraph in H, then define f (C) as follows: # degree 2’s in C 1 2 3 4 5 6 7 8 9 f (C) 1 1 1 2 2 3 4 If R ∈ B then f (R) =    2 R is a component in H 1 R is intersected by 1 edge R is intersected by ≥ 2 edges . Let ρ = max

Q∈Q f (Q) over all families of vertex-disjoint

subgraphs Q in H (F(Q) = 0 if Q ∈ B ∪ SB1).

Anders Yeo Transversals in hypergraphs

Progress on the Tuza-Vestergaard Conjecture

Using the above complicated definition of ρ we can prove that 20τ(H) ≤ 2n + 6m + ρ. This implies the Tuza-Vestergaard Conjecture, as follows. When H is 3-regular then 3n =

v∈V d(v) = 6m.

Also, ρ = 0 as f (Q) = 0 if all vertices of Q have degree three. 20τ(H) ≤ 2n + 6m + ρ = 2n + 6m = 2n + 3n = 5n So, τ(H) ≤ n/4. QED.

Anders Yeo Transversals in hypergraphs

Progress on the Tuza-Vestergaard Conjecture

Using the above complicated definition of ρ we can prove that 20τ(H) ≤ 2n + 6m + ρ. This implies the Tuza-Vestergaard Conjecture, as follows. When H is 3-regular then 3n =

v∈V d(v) = 6m.

Also, ρ = 0 as f (Q) = 0 if all vertices of Q have degree three. 20τ(H) ≤ 2n + 6m + ρ = 2n + 6m = 2n + 3n = 5n So, τ(H) ≤ n/4. QED.

Anders Yeo Transversals in hypergraphs

Progress on the Tuza-Vestergaard Conjecture

Using the above complicated definition of ρ we can prove that 20τ(H) ≤ 2n + 6m + ρ. This implies the Tuza-Vestergaard Conjecture, as follows. When H is 3-regular then 3n =

v∈V d(v) = 6m.

Also, ρ = 0 as f (Q) = 0 if all vertices of Q have degree three. 20τ(H) ≤ 2n + 6m + ρ = 2n + 6m = 2n + 3n = 5n So, τ(H) ≤ n/4. QED.

Anders Yeo Transversals in hypergraphs

Progress on the Tuza-Vestergaard Conjecture

Using the above complicated definition of ρ we can prove that 20τ(H) ≤ 2n + 6m + ρ. This implies the Tuza-Vestergaard Conjecture, as follows. When H is 3-regular then 3n =

v∈V d(v) = 6m.

Also, ρ = 0 as f (Q) = 0 if all vertices of Q have degree three. 20τ(H) ≤ 2n + 6m + ρ = 2n + 6m = 2n + 3n = 5n So, τ(H) ≤ n/4. QED.

Anders Yeo Transversals in hypergraphs

Progress on the Tuza-Vestergaard Conjecture

Using the above complicated definition of ρ we can prove that 20τ(H) ≤ 2n + 6m + ρ. This implies the Tuza-Vestergaard Conjecture, as follows. When H is 3-regular then 3n =

v∈V d(v) = 6m.

Also, ρ = 0 as f (Q) = 0 if all vertices of Q have degree three. 20τ(H) ≤ 2n + 6m + ρ = 2n + 6m = 2n + 3n = 5n So, τ(H) ≤ n/4. QED.

Anders Yeo Transversals in hypergraphs

Progress on a conjecture for linear 4-uniform hypergraphs

Theorem: If H is 2-uniform then τ(H) ≤ n+m

3 .

Theorem: If H is 3-uniform then τ(H) ≤ n+m

4 .

The unique simple k-uniform hypergraph on k + 1 vertices with k + 1 edges has transversal number 2. So the above theorems are tight (also due to the single edge). It is not true that τ(H) ≤ n+m

5

for 4-uniform hypergraphs H. What happens if we look at linear hypergraphs?

Anders Yeo Transversals in hypergraphs

Progress on a conjecture for linear 4-uniform hypergraphs

Theorem: If H is 2-uniform then τ(H) ≤ n+m

3 .

Theorem: If H is 3-uniform then τ(H) ≤ n+m

4 .

The unique simple k-uniform hypergraph on k + 1 vertices with k + 1 edges has transversal number 2. So the above theorems are tight (also due to the single edge). It is not true that τ(H) ≤ n+m

5

for 4-uniform hypergraphs H. What happens if we look at linear hypergraphs?

Anders Yeo Transversals in hypergraphs

Progress on a conjecture for linear 4-uniform hypergraphs

Theorem: If H is 2-uniform then τ(H) ≤ n+m

3 .

Theorem: If H is 3-uniform then τ(H) ≤ n+m

4 .

The unique simple k-uniform hypergraph on k + 1 vertices with k + 1 edges has transversal number 2. So the above theorems are tight (also due to the single edge). It is not true that τ(H) ≤ n+m

5

for 4-uniform hypergraphs H. What happens if we look at linear hypergraphs?

Anders Yeo Transversals in hypergraphs

Progress on a conjecture for linear 4-uniform hypergraphs

Theorem: If H is 2-uniform then τ(H) ≤ n+m

3 .

Theorem: If H is 3-uniform then τ(H) ≤ n+m

4 .

The unique simple k-uniform hypergraph on k + 1 vertices with k + 1 edges has transversal number 2. So the above theorems are tight (also due to the single edge). It is not true that τ(H) ≤ n+m

5

for 4-uniform hypergraphs H. What happens if we look at linear hypergraphs?

Anders Yeo Transversals in hypergraphs

Progress on a conjecture for linear 4-uniform hypergraphs

Theorem: If H is 2-uniform then τ(H) ≤ n+m

3 .

Theorem: If H is 3-uniform then τ(H) ≤ n+m

4 .

The unique simple k-uniform hypergraph on k + 1 vertices with k + 1 edges has transversal number 2. So the above theorems are tight (also due to the single edge). It is not true that τ(H) ≤ n+m

5

for 4-uniform hypergraphs H. What happens if we look at linear hypergraphs?

Anders Yeo Transversals in hypergraphs

Progress on a conjecture for linear 4-uniform hypergraphs

Theorem: If H is 2-uniform then τ(H) ≤ n+m

3 .

Theorem: If H is 3-uniform then τ(H) ≤ n+m

4 .

The unique simple k-uniform hypergraph on k + 1 vertices with k + 1 edges has transversal number 2. So the above theorems are tight (also due to the single edge). It is not true that τ(H) ≤ n+m

5

for 4-uniform hypergraphs H. What happens if we look at linear hypergraphs?

Anders Yeo Transversals in hypergraphs

Progress on a conjecture for linear 4-uniform hypergraphs

Surprisingly little is know about best possible bounds for linear k-uniform hypergraphs. Question: Is it true that τ(H) ≤ n+m

k+1 for all linear k-uniform

hypergraphs? The answer is yes for k = 2 and k = 3 (even if we do not require linearity). What about k = 4?

Anders Yeo Transversals in hypergraphs

Progress on a conjecture for linear 4-uniform hypergraphs

Surprisingly little is know about best possible bounds for linear k-uniform hypergraphs. Question: Is it true that τ(H) ≤ n+m

k+1 for all linear k-uniform

hypergraphs? The answer is yes for k = 2 and k = 3 (even if we do not require linearity). What about k = 4?

Anders Yeo Transversals in hypergraphs

Progress on a conjecture for linear 4-uniform hypergraphs

Surprisingly little is know about best possible bounds for linear k-uniform hypergraphs. Question: Is it true that τ(H) ≤ n+m

k+1 for all linear k-uniform

hypergraphs? The answer is yes for k = 2 and k = 3 (even if we do not require linearity). What about k = 4?

Anders Yeo Transversals in hypergraphs

Progress on a conjecture for linear 4-uniform hypergraphs

Surprisingly little is know about best possible bounds for linear k-uniform hypergraphs. Question: Is it true that τ(H) ≤ n+m

k+1 for all linear k-uniform

hypergraphs? The answer is yes for k = 2 and k = 3 (even if we do not require linearity). What about k = 4?

Anders Yeo Transversals in hypergraphs

Progress on a conjecture for linear 4-uniform hypergraphs

Conjecture: τ(H) ≤ n+m

k+1 for all 4-uniform hypergraphs.

We believe we can prove the above conjecture (we will write up the result next week!). This result would also imply that the following conjecture holds for k = 4 (it is known to hold for k = 2, 3). Conjecture: For all k ≥ 2, if H is a k-uniform linear hypergraph then τ(H) ≤ |V (H)|

k

+ |E(H)|

6

. Does the above conjecture look familiar? Substitute ”linear” with ”∆ ≤ 3”...

Anders Yeo Transversals in hypergraphs

Progress on a conjecture for linear 4-uniform hypergraphs

Conjecture: τ(H) ≤ n+m

k+1 for all 4-uniform hypergraphs.

We believe we can prove the above conjecture (we will write up the result next week!). This result would also imply that the following conjecture holds for k = 4 (it is known to hold for k = 2, 3). Conjecture: For all k ≥ 2, if H is a k-uniform linear hypergraph then τ(H) ≤ |V (H)|

k

+ |E(H)|

6

. Does the above conjecture look familiar? Substitute ”linear” with ”∆ ≤ 3”...

Anders Yeo Transversals in hypergraphs

Progress on a conjecture for linear 4-uniform hypergraphs

Conjecture: τ(H) ≤ n+m

k+1 for all 4-uniform hypergraphs.

We believe we can prove the above conjecture (we will write up the result next week!). This result would also imply that the following conjecture holds for k = 4 (it is known to hold for k = 2, 3). Conjecture: For all k ≥ 2, if H is a k-uniform linear hypergraph then τ(H) ≤ |V (H)|

k

+ |E(H)|

6

. Does the above conjecture look familiar? Substitute ”linear” with ”∆ ≤ 3”...

Anders Yeo Transversals in hypergraphs

Progress on a conjecture for linear 4-uniform hypergraphs

Conjecture: τ(H) ≤ n+m

k+1 for all 4-uniform hypergraphs.

We believe we can prove the above conjecture (we will write up the result next week!). This result would also imply that the following conjecture holds for k = 4 (it is known to hold for k = 2, 3). Conjecture: For all k ≥ 2, if H is a k-uniform linear hypergraph then τ(H) ≤ |V (H)|

k

+ |E(H)|

6

. Does the above conjecture look familiar? Substitute ”linear” with ”∆ ≤ 3”...

Anders Yeo Transversals in hypergraphs

Progress on a conjecture for linear 4-uniform hypergraphs

Conjecture: τ(H) ≤ n+m

k+1 for all 4-uniform hypergraphs.

We believe we can prove the above conjecture (we will write up the result next week!). This result would also imply that the following conjecture holds for k = 4 (it is known to hold for k = 2, 3). Conjecture: For all k ≥ 2, if H is a k-uniform linear hypergraph then τ(H) ≤ |V (H)|

k

+ |E(H)|

6

. Does the above conjecture look familiar? Substitute ”linear” with ”∆ ≤ 3”...

Anders Yeo Transversals in hypergraphs

Progress on a conjecture for linear 4-uniform hypergraphs

Conjecture: τ(H) ≤ n+m

k+1 for all 4-uniform hypergraphs.

We believe we can prove the above conjecture (we will write up the result next week!). This result would also imply that the following conjecture holds for k = 4 (it is known to hold for k = 2, 3). Conjecture: For all k ≥ 2, if H is a k-uniform linear hypergraph then τ(H) ≤ |V (H)|

k

+ |E(H)|

6

. Does the above conjecture look familiar? Substitute ”linear” with ”∆ ≤ 3”...

Anders Yeo Transversals in hypergraphs

Progress on a conjecture for linear 4-uniform hypergraphs

How do we prove that τ(H) ≤ n+m

k+1 for all 4-uniform hypergraphs?

Again we need to prove a much stronger result. And we need to introduce a new concept called deficiency, def (H),

• f a hypergraph H.

Theorem (stronger result): If H is a 4-uniform linear hypergraph, with ∆(H) ≤ 3, then 45τ(H) ≤ 6|V (H)| + 13|E(H)| + def (H).

Anders Yeo Transversals in hypergraphs

Progress on a conjecture for linear 4-uniform hypergraphs

How do we prove that τ(H) ≤ n+m

k+1 for all 4-uniform hypergraphs?

Again we need to prove a much stronger result. And we need to introduce a new concept called deficiency, def (H),

• f a hypergraph H.

Theorem (stronger result): If H is a 4-uniform linear hypergraph, with ∆(H) ≤ 3, then 45τ(H) ≤ 6|V (H)| + 13|E(H)| + def (H).

Anders Yeo Transversals in hypergraphs

Progress on a conjecture for linear 4-uniform hypergraphs

How do we prove that τ(H) ≤ n+m

k+1 for all 4-uniform hypergraphs?

Again we need to prove a much stronger result. And we need to introduce a new concept called deficiency, def (H),

• f a hypergraph H.

Theorem (stronger result): If H is a 4-uniform linear hypergraph, with ∆(H) ≤ 3, then 45τ(H) ≤ 6|V (H)| + 13|E(H)| + def (H).

Anders Yeo Transversals in hypergraphs

Progress on a conjecture for linear 4-uniform hypergraphs

How do we prove that τ(H) ≤ n+m

k+1 for all 4-uniform hypergraphs?

Again we need to prove a much stronger result. And we need to introduce a new concept called deficiency, def (H),

• f a hypergraph H.

Theorem (stronger result): If H is a 4-uniform linear hypergraph, with ∆(H) ≤ 3, then 45τ(H) ≤ 6|V (H)| + 13|E(H)| + def (H).

Anders Yeo Transversals in hypergraphs

Progress on a conjecture for linear 4-uniform hypergraphs

The following 16 ”bad” hypergraphs add to the deficiency.

Anders Yeo Transversals in hypergraphs

Progress on a conjecture for linear 4-uniform hypergraphs

Let g(R) be the amount that the graph G is short in 45τ(R) ≤ 6|V (R)| + 13|E(R)|. Let R be a vertex disjoint collection of bad subhypergraphs and let e(R) be the number of edges intersecting these in H. The deficiency of H is the maximum possible value of (

R∈R g(R)) − 13e(R) over all R.

def (H) = 4 × 8 − 2 × 13 = 6.

Anders Yeo Transversals in hypergraphs

Progress on a conjecture for linear 4-uniform hypergraphs

Let g(R) be the amount that the graph G is short in 45τ(R) ≤ 6|V (R)| + 13|E(R)|. Let R be a vertex disjoint collection of bad subhypergraphs and let e(R) be the number of edges intersecting these in H. The deficiency of H is the maximum possible value of (

R∈R g(R)) − 13e(R) over all R.

def (H) = 4 × 8 − 2 × 13 = 6.

Anders Yeo Transversals in hypergraphs

Progress on a conjecture for linear 4-uniform hypergraphs

Let g(R) be the amount that the graph G is short in 45τ(R) ≤ 6|V (R)| + 13|E(R)|. Let R be a vertex disjoint collection of bad subhypergraphs and let e(R) be the number of edges intersecting these in H. The deficiency of H is the maximum possible value of (

R∈R g(R)) − 13e(R) over all R.

def (H) = 4 × 8 − 2 × 13 = 6.

Anders Yeo Transversals in hypergraphs

Progress on a conjecture for linear 4-uniform hypergraphs

Let g(R) be the amount that the graph G is short in 45τ(R) ≤ 6|V (R)| + 13|E(R)|. Let R be a vertex disjoint collection of bad subhypergraphs and let e(R) be the number of edges intersecting these in H. The deficiency of H is the maximum possible value of (

R∈R g(R)) − 13e(R) over all R.

def (H) = 4 × 8 − 2 × 13 = 6.

Anders Yeo Transversals in hypergraphs

Progress on a conjecture for linear 4-uniform hypergraphs

Let g(R) be the amount that the graph G is short in 45τ(R) ≤ 6|V (R)| + 13|E(R)|. Let R be a vertex disjoint collection of bad subhypergraphs and let e(R) be the number of edges intersecting these in H. The deficiency of H is the maximum possible value of (

R∈R g(R)) − 13e(R) over all R.

def (H) = 4 × 8 − 2 × 13 = 6.

Anders Yeo Transversals in hypergraphs

Progress on a conjecture for linear 4-uniform hypergraphs

Let g(R) be the amount that the graph G is short in 45τ(R) ≤ 6|V (R)| + 13|E(R)|. Let R be a vertex disjoint collection of bad subhypergraphs and let e(R) be the number of edges intersecting these in H. The deficiency of H is the maximum possible value of (

R∈R g(R)) − 13e(R) over all R.

def (H) = 4 × 8 − 2 × 13 = 6.

Anders Yeo Transversals in hypergraphs

Consequences

Apart from proving the two conjectures stated, the following also follows. Theorem? Let H be a 3-regular 4-uniform hypergraph on n

• vertices. If H is linear, then τ(H) ≤ 7n/20.

Theorem? If G is a graph of order n, with no 4-cycles and with δ(G) ≥ 4, then γt(G) ≤ 2n/5.

Anders Yeo Transversals in hypergraphs

Consequences

Apart from proving the two conjectures stated, the following also follows. Theorem? Let H be a 3-regular 4-uniform hypergraph on n

• vertices. If H is linear, then τ(H) ≤ 7n/20.

Theorem? If G is a graph of order n, with no 4-cycles and with δ(G) ≥ 4, then γt(G) ≤ 2n/5.

Anders Yeo Transversals in hypergraphs

Consequences

Apart from proving the two conjectures stated, the following also follows. Theorem? Let H be a 3-regular 4-uniform hypergraph on n

• vertices. If H is linear, then τ(H) ≤ 7n/20.

Theorem? If G is a graph of order n, with no 4-cycles and with δ(G) ≥ 4, then γt(G) ≤ 2n/5.

Anders Yeo Transversals in hypergraphs

The End.

The End! Thank you. Any Questions?

Anders Yeo Transversals in hypergraphs

The End.

The End! Thank you. Any Questions?

Anders Yeo Transversals in hypergraphs