[PPT] - Reduction of the Berge-Fulkerson Conjecture to cyclically PowerPoint Presentation

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Reduction of the Berge-Fulkerson Conjecture to cyclically 5-edge-connected snarks

Giuseppe Mazzuoccolo

University of Verona, Italy GGTW 2019

joint work with Edita M´ aˇ cajov´ a (Comenius University, Bratislava)

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 1 / 25

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Berge-Fulkerson Conjecture

Conjecture (Berge-Fulkerson, 1971)

Every bridgeless cubic graph contains a family of SIX perfect matchings that together cover each edge exactly twice.

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 2 / 25

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Berge-Fulkerson Conjecture

Conjecture (Berge-Fulkerson, 1971)

Every bridgeless cubic graph contains a family of SIX perfect matchings that together cover each edge exactly twice.

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 2 / 25

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Introduction

Berge-Fulkerson Conjecture (1971)

Every bridgeless cubic graph contains a family of SIX perfect matchings that together cover each edge exactly twice.

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 3 / 25

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Introduction

Berge-Fulkerson Conjecture (1971)

Every bridgeless cubic graph contains a family of SIX perfect matchings that together cover each edge exactly twice. trivial for 3-edge-colourable cubic graphs

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 3 / 25

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Introduction

Berge-Fulkerson Conjecture (1971)

Every bridgeless cubic graph contains a family of SIX perfect matchings that together cover each edge exactly twice. trivial for 3-edge-colourable cubic graphs hard for bridgeless cubic graphs which are not 3-edge-colourable (these graphs were named SNARKS by Martin Gardner).

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 3 / 25

SLIDE 7

Introduction

Berge-Fulkerson Conjecture (1971)

Every bridgeless cubic graph contains a family of SIX perfect matchings that together cover each edge exactly twice. trivial for 3-edge-colourable cubic graphs hard for bridgeless cubic graphs which are not 3-edge-colourable (these graphs were named SNARKS by Martin Gardner). Do we need to require a graph to be bridgeless?

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 3 / 25

SLIDE 8

Introduction

Berge-Fulkerson Conjecture (1971)

Every bridgeless cubic graph contains a family of SIX perfect matchings that together cover each edge exactly twice. trivial for 3-edge-colourable cubic graphs hard for bridgeless cubic graphs which are not 3-edge-colourable (these graphs were named SNARKS by Martin Gardner). Do we need to require a graph to be bridgeless?

◮ YES! (a bridge in a cubic graph belongs to every perfect matching) Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 3 / 25

SLIDE 9

Introduction

Berge-Fulkerson Conjecture (1971)

Every bridgeless cubic graph contains a family of SIX perfect matchings that together cover each edge exactly twice. trivial for 3-edge-colourable cubic graphs hard for bridgeless cubic graphs which are not 3-edge-colourable (these graphs were named SNARKS by Martin Gardner). Do we need to require a graph to be bridgeless?

◮ YES! (a bridge in a cubic graph belongs to every perfect matching)

ALTERNATIVE FORMULATION: if we double edges in a bridgeless cubic graph, we obtain a 6-edge-colourable 6-regular multigraph

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 3 / 25

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Cyclic connectivity and oddness

Cyclic connectivity is the smallest number of edges which have to be removed in order to obtain at least two components containing cycles

✁✁✁✁✁✁ ✂✁✂✁✂✁✂✁✂✁✂✁✂ ✄✁✄✁✄✁✄✁✄✁✄✁✄ ☎✁☎✁☎✁☎✁☎✁☎✁☎ ✆✁✆✁✆✁✆✁✆✁✆✁✆ ✝✁✝✁✝✁✝✁✝✁✝✁✝ ✞✁✞✁✞✁✞✁✞✁✞✁✞ ✟✁✟✁✟✁✟✁✟✁✟✁✟

k edges

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Cyclic connectivity and oddness

Cyclic connectivity is the smallest number of edges which have to be removed in order to obtain at least two components containing cycles

✁✁✁✁✁✁ ✂✁✂✁✂✁✂✁✂✁✂✁✂ ✄✁✄✁✄✁✄✁✄✁✄✁✄ ☎✁☎✁☎✁☎✁☎✁☎✁☎ ✆✁✆✁✆✁✆✁✆✁✆✁✆ ✝✁✝✁✝✁✝✁✝✁✝✁✝ ✞✁✞✁✞✁✞✁✞✁✞✁✞ ✟✁✟✁✟✁✟✁✟✁✟✁✟

k edges

Conjecture (Jaeger, Swart’80)

There is no snark with cyclic connectivity greater than 6.

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 4 / 25

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Cyclic connectivity and oddness

Cyclic connectivity is the smallest number of edges which have to be removed in order to obtain at least two components containing cycles

✁✁✁✁✁✁ ✂✁✂✁✂✁✂✁✂✁✂✁✂ ✄✁✄✁✄✁✄✁✄✁✄✁✄ ☎✁☎✁☎✁☎✁☎✁☎✁☎ ✆✁✆✁✆✁✆✁✆✁✆✁✆ ✝✁✝✁✝✁✝✁✝✁✝✁✝ ✞✁✞✁✞✁✞✁✞✁✞✁✞ ✟✁✟✁✟✁✟✁✟✁✟✁✟

k edges

Oddness ω(G) of a bridgeless cubic graph G is the smallest number of odd cycles in a 2-factor of G.

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 4 / 25

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Cyclic connectivity and oddness

Cyclic connectivity is the smallest number of edges which have to be removed in order to obtain at least two components containing cycles

✁✁✁✁✁✁ ✂✁✂✁✂✁✂✁✂✁✂✁✂ ✄✁✄✁✄✁✄✁✄✁✄✁✄ ☎✁☎✁☎✁☎✁☎✁☎✁☎ ✆✁✆✁✆✁✆✁✆✁✆✁✆ ✝✁✝✁✝✁✝✁✝✁✝✁✝ ✞✁✞✁✞✁✞✁✞✁✞✁✞ ✟✁✟✁✟✁✟✁✟✁✟✁✟

k edges

Oddness ω(G) of a bridgeless cubic graph G is the smallest number of odd cycles in a 2-factor of G. ω(G) = 0 ⇔ G is 3-edge-colourable

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 4 / 25

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Possible Minimal Counterexamples to some Outstanding Conjectures

5−flow 5−cycle double Conjecture conj. girth cyclic connectivity

ddness

≥ 6

[GM, Steffen]

≥ 6

[Huck]

≥ 2 ≥ 4 ≥ 4

[Kochol]

≥ 6 ≥ 11

[Kochol]

≥ 12

[Huck]

≥ 5 Conjecture cover C. Berge-Fulkerson

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 5 / 25

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Possible Minimal Counterexamples to some Outstanding Conjectures

5−flow 5−cycle double Conjecture conj. girth cyclic connectivity

ddness

≥ 6

[GM, Steffen]

≥ 6

[Huck]

≥ 2 ≥ 4

[Kochol]

≥ 6 ≥ 11

[Kochol]

≥ 12

[Huck]

≥ 5 Conjecture cover C. Berge-Fulkerson ≥ 5

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 5 / 25

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BF-colourings

Let G be a bridgeless cubic graph. Consider six perfect matchings of G, say {M1, M2, M3, M4, M5, M6}, such that every edge of G belongs to exactly two of them.

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 6 / 25

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BF-colourings

Let G be a bridgeless cubic graph. Consider six perfect matchings of G, say {M1, M2, M3, M4, M5, M6}, such that every edge of G belongs to exactly two of them. These perfect matchings induce a map φ : E(G) → { 2-subsets of {1, 2, 3, 4, 5, 6}} φ(e) = {i, j}, i = j and φ(e) ∩ φ(f ) = ∅ for all pairs of incident edges e, f .

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 6 / 25

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BF-colourings

Let G be a bridgeless cubic graph. Consider six perfect matchings of G, say {M1, M2, M3, M4, M5, M6}, such that every edge of G belongs to exactly two of them. These perfect matchings induce a map φ : E(G) → { 2-subsets of {1, 2, 3, 4, 5, 6}} φ(e) = {i, j}, i = j and φ(e) ∩ φ(f ) = ∅ for all pairs of incident edges e, f . We say that φ is a BF-colouring of G.

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 6 / 25

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BF-colourings of 4-poles

16 35 15 36

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BF-colourings of 4-poles

16 35 15 36 16 35 15 36

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 7 / 25

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BF-colourings of 4-poles

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 7 / 25

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BF-colourings of 4-poles

?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ??

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 7 / 25

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BF-colourings of 4-poles

?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ??

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 7 / 25

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BF-colourings of 4-poles

?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ??

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 7 / 25

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BF-colourings of 4-poles

?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ??

12 13 12 13

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4-edge-cut colourings

There are exactly 4 types of possible partions of the 4 dangling edges along two disjoint perfect matchings:

A T4 T3 T2

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”Splitting” of a BF-colouring of a 4-edge-cut

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 9 / 25

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”Splitting” of a BF-colouring of a 4-edge-cut

2 2 1 1 1 2 2 1 AA

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”Splitting” of a BF-colouring of a 4-edge-cut

2 2 1 1 1 2 2 1 3 1 1 1 1 2 2 3 AA AT2

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”Splitting” of a BF-colouring of a 4-edge-cut

2 2 1 1 1 2 2 1 3 1 1 1 1 2 2 3 1 1 AA AT2

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 9 / 25

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”Splitting” of a BF-colouring of a 4-edge-cut

2 2 1 1 1 2 2 1 3 1 1 1 1 2 2 3 1 1 2 3 AA AT2

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 9 / 25

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”Splitting” of a BF-colouring of a 4-edge-cut

2 2 1 1 1 2 2 1 3 1 1 1 1 2 2 3 1 1 2 3 2 3 AA AT2

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 9 / 25

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”Splitting” of a BF-colouring of a 4-edge-cut

2 2 1 1 1 2 2 1 3 1 1 1 1 2 2 3 1 1 2 3 2 3 4 4 AA AT2

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 9 / 25

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”Splitting” of a BF-colouring of a 4-edge-cut

2 2 1 1 1 2 2 1 3 1 1 1 1 2 2 3 1 1 2 3 2 3 4 4 1 2 3 1 4 2 3 4 AA AT2 T2T3

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 9 / 25

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”Splitting” of a BF-colouring of a 4-edge-cut

2 2 1 1 1 2 2 1 3 1 1 1 1 2 2 3 1 1 2 3 2 3 4 4 1 2 3 1 4 2 3 4 AA AT2 T2T3

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 9 / 25

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”Splitting” of a BF-colouring of a 4-edge-cut

2 2 1 1 1 2 2 1 3 1 1 1 1 2 2 3 1 1 2 3 2 3 4 4 1 2 3 1 4 2 3 4 AA AT2 T2T3

there exist 4

2

+ 4 = 10 types of BF-colourings of a 4-edge-cut

{AA, AT2, AT3, AT4, T2T2, T2T3, T2T4, T3T3, T3T4, T4T4}

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 9 / 25

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”Splitting” of a BF-colouring of a 4-edge-cut

2 2 1 1 1 2 2 1 3 1 1 1 1 2 2 3 1 1 2 3 2 3 4 4 1 2 3 1 4 2 3 4 AA AT2 T2T3

there exist 4

2

+ 4 = 10 types of BF-colourings of a 4-edge-cut

{AA, AT2, AT3, AT4, T2T2, T2T3, T2T4, T3T3, T3T4, T4T4} we can associate to every 4-pole one of the 210 possible subsets of types of colouring, BUT

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 9 / 25

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”Splitting” of a BF-colouring of a 4-edge-cut

2 2 1 1 1 2 2 1 3 1 1 1 1 2 2 3 1 1 2 3 2 3 4 4 1 2 3 1 4 2 3 4 AA AT2 T2T3

there exist 4

2

+ 4 = 10 types of BF-colourings of a 4-edge-cut

{AA, AT2, AT3, AT4, T2T2, T2T3, T2T4, T3T3, T3T4, T4T4} we can associate to every 4-pole one of the 210 possible subsets of types of colouring, BUT not all of them are achievable...

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 9 / 25

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Kempe chains

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 10 / 25

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Kempe chains

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 10 / 25

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Kempe chains

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 10 / 25

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Kempe chains

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 10 / 25

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Kempe chains

∨ ∨

12 12 12

AT2 AT3 AT4

12 AA

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Kempe chains

AND AND AND AND AND AND

∨ ∨ 24 24 13 ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ 12 13 13 12 12 12 13 12 13 13 13 12 14 23 24 14 24 23 13 24 24 ∨ ∨ ∨ ∨ ∨ ∨ 24 13 24 24 14 23

AT2 AT3 AT4 T2T2 T2T3 T2T4 AA AT3 AT4 AA AT1 AT2 T2T4 T3T4 T4T4

AT2 AT3 AT4 T2T2 T2T3 T2T4

AT2 T2T3 T2T4 AT2 T2T3 T2T4 AT3 T3T3 T3T4 AT2 T2T2 T2T3 AT4 T3T4 T4T4 AA AT2 AT4 T2T3 T3T3 T3T4 AT3 T2T3 T3T4 AT3 T2T3 T3T3 AT4 T2T4 T4T4 AT4 T2T4 T3T4

T4T4

13 12 12

AA

12 12 13 13 13 13 13

T3T3 T3T4

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Graph of BF-colourings

each 4-pole corresponds to a subgraph of M according to its admissible BF-colourings

T4 T2 A T3

T3T4 AT3 AT4 AT2 T2T2 T4T4 T2T2 AA T2T4 T3T3

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4-pole → a subgraph of M

T4 T2 A T3

24 24 13 24 14 23 24 13 24 T3T4 AT3 AT4 AT2 T2T2 T4T4 T2T2 AA T2T4 T3T3 13 13 13

56 56 56

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4-pole → a subgraph of M

T4 T2 A T3

24 24 13 24 14 23 24 13 24 T3T4 AT3 AT4 AT2 T2T2 T4T4 T2T2 AA T2T4 T3T3

T4T4 T3T4 T3T3

13 13 13

56 56 56

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Acyclic 4-poles

There are only SIX different acyclic 4-poles. In each of them, the admissible BF-colourings correspond to one of the SIX dumbbell subgraphs

f M.

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Main result

Theorem

A smallest possible counterexample to the Berge-Fulkerson conjecture is cyclically 5-edge-connected. Sketch of the proof.

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Main result

Theorem

A smallest possible counterexample to the Berge-Fulkerson conjecture is cyclically 5-edge-connected. Sketch of the proof. a smallest counterexample is cyclically 4-edge-connected

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 16 / 25

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Main result

Theorem

A smallest possible counterexample to the Berge-Fulkerson conjecture is cyclically 5-edge-connected. Sketch of the proof. a smallest counterexample is cyclically 4-edge-connected assume that G is a smallest counterexample and that G contains a cycle separating 4-edge-cut S S G1 G2 M1 M2

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 16 / 25

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Main result

Theorem

A smallest possible counterexample to the Berge-Fulkerson conjecture is cyclically 5-edge-connected. Sketch of the proof. a smallest counterexample is cyclically 4-edge-connected assume that G is a smallest counterexample and that G contains a cycle separating 4-edge-cut S S G1 G2 M1 M2

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 16 / 25

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Main result

Theorem

A smallest possible counterexample to the Berge-Fulkerson conjecture is cyclically 5-edge-connected. Sketch of the proof. a smallest counterexample is cyclically 4-edge-connected assume that G is a smallest counterexample and that G contains a cycle separating 4-edge-cut S S G1 G2 M1 M2 subgraphs of M

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Sketch of the proof

M1 and M2 are edge-disjoint

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 17 / 25

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Sketch of the proof

M1 and M2 are edge-disjoint

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 17 / 25

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Sketch of the proof

M1 and M2 are edge-disjoint

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 17 / 25

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Sketch of the proof

M1 and M2 are edge-disjoint

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 17 / 25

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Sketch of the proof

M1 and M2 are edge-disjoint both G1 and G2 admit a BF-colouring, otherwise we have a contradiction with the minimality of G, therefore

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 17 / 25

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Sketch of the proof

M1 and M2 are edge-disjoint both G1 and G2 admit a BF-colouring, otherwise we have a contradiction with the minimality of G, therefore

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 17 / 25

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Sketch of the proof

M1 and M2 are edge-disjoint both G1 and G2 admit a BF-colouring, otherwise we have a contradiction with the minimality of G, therefore neither Mi nor Mi contains a dumbbell subgraph

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No subgraph of Mi or Mi

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No subgraph of Mi or Mi

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No subgraph of Mi or Mi

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No subgraph of Mi or Mi

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No subgraph of Mi or Mi

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No subgraph of Mi or Mi

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No subgraph of Mi or Mi

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No subgraph of Mi or Mi

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No subgraph of Mi or Mi

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No subgraph of Mi or Mi

SMALLER COUNTEREXAMPLE!

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Sketch of the proof

M1 and M2 are edge-disjoint both G1 and G2 admit a BF-colouring, otherwise we have a contradiction with the minimality of G, therefore neither Mi nor Mi contains a subgraph isomorphic to

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SLIDE 72

Sketch of the proof

M1 and M2 are edge-disjoint both G1 and G2 admit a BF-colouring, otherwise we have a contradiction with the minimality of G, therefore neither Mi nor Mi contains a subgraph isomorphic to no vertices of degree 1 in M1 nor M2 (Kempe chains)

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 19 / 25

SLIDE 73

Sketch of the proof

M1 and M2 are edge-disjoint both G1 and G2 admit a BF-colouring, otherwise we have a contradiction with the minimality of G, therefore neither Mi nor Mi contains a subgraph isomorphic to no vertices of degree 1 in M1 nor M2 (Kempe chains) no vertices of degree 2 in M1 nor M2 incident with a loop (Kempe chains) further (and last) forbidden configuration....

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a further forbidden configuration....

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 20 / 25

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a further forbidden configuration....

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 20 / 25

SLIDE 76

Sketch of the proof

M1 and M2 are edge-disjoint both G1 and G2 admit a BF-colouring, otherwise we have a contradiction with the minimality of G, therefore neither Mi nor Mi contains a subgraph isomorphic to no vertices of degree 1 in M1 nor M2 (Kempe chains) no vertices of degree 2 in M1 nor M2 incident with a loop (Kempe chains) further forbidden configuration....

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 21 / 25

SLIDE 77

Sketch of the proof

M1 and M2 are edge-disjoint both G1 and G2 admit a BF-colouring, otherwise we have a contradiction with the minimality of G, therefore neither Mi nor Mi contains a subgraph isomorphic to no vertices of degree 1 in M1 nor M2 (Kempe chains) no vertices of degree 2 in M1 nor M2 incident with a loop (Kempe chains) further forbidden configuration.... ...CONTRADICTION!

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 21 / 25

SLIDE 78

WORK IN PROGRESS: 5-edge-cuts

56 types of colourings

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 22 / 25

SLIDE 79

WORK IN PROGRESS: 5-edge-cuts

56 types of colourings 256 subsets

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 22 / 25

SLIDE 80

WORK IN PROGRESS: 5-edge-cuts

56 types of colourings 256 subsets with the help of a computer we identified the subsets that are

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 22 / 25

SLIDE 81

WORK IN PROGRESS: 5-edge-cuts

56 types of colourings 256 subsets with the help of a computer we identified the subsets that are

◮ closed under 1 and 2 Kempe switches Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 22 / 25

SLIDE 82

WORK IN PROGRESS: 5-edge-cuts

56 types of colourings 256 subsets with the help of a computer we identified the subsets that are

◮ closed under 1 and 2 Kempe switches ◮ do not contain a subsets of colourings corresponding to an acyclic

5-pole

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 22 / 25

SLIDE 83

WORK IN PROGRESS: 5-edge-cuts

56 types of colourings 256 subsets with the help of a computer we identified the subsets that are

◮ closed under 1 and 2 Kempe switches ◮ do not contain a subsets of colourings corresponding to an acyclic

5-pole

◮ their complement does not contain a subsets of colourings

corresponding to an acyclic 5-pole

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 22 / 25

SLIDE 84

WORK IN PROGRESS: 5-edge-cuts

56 types of colourings 256 subsets with the help of a computer we identified the subsets that are

◮ closed under 1 and 2 Kempe switches ◮ do not contain a subsets of colourings corresponding to an acyclic

5-pole

◮ their complement does not contain a subsets of colourings

corresponding to an acyclic 5-pole

◮ have in complement one of such sets Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 22 / 25

SLIDE 85

WORK IN PROGRESS: 5-edge-cuts

56 types of colourings 256 subsets with the help of a computer we identified the subsets that are

◮ closed under 1 and 2 Kempe switches ◮ do not contain a subsets of colourings corresponding to an acyclic

5-pole

◮ their complement does not contain a subsets of colourings

corresponding to an acyclic 5-pole

◮ have in complement one of such sets

13 pairs left of sets of colourings

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 22 / 25

SLIDE 86

https://combinatorics2020.unibs.it

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 23 / 25

SLIDE 87

List of plenary speakers

Herivelto BORGES - University of San Paulo (Brasil) Bence CSAJBOK - Eotvos Lorand University (Hungary) Nicola DURANTE - University of Naples ”Federico II” (Italy) Michel LAVRAUW - Sabanci University (Turkey) Patric R. J. OSTERGARD - Aalto University (Finland) Tomaz PISANSKI - Primorska University (Slovenia) Violet R. SYROTIUK - Arizona State University (USA) Ian WANLESS - Monash University (Australia)

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 24 / 25

SLIDE 88

Thank you for your attention!

Giuseppe Mazzuoccolo (Verona University) Berge-Fulkerson conjecture August 12-14, 2019 25 / 25