On Vizings edge-colouring question Marthe Bonamy July 28, 2020 - - PowerPoint PPT Presentation

On Vizing’s edge-colouring question

Marthe Bonamy July 28, 2020

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Edge colouring

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Edge colouring

χ′: Minimum number of colors to ensure that a b ⇒ a = b.

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Edge colouring

χ′: Minimum number of colors to ensure that a b ⇒ a = b. ∆: Maximum degree of the graph. ∆ ≤ χ′

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Vizing’s theorem and Kempe equivalence

Theorem (Vizing ’64) For any graph G, ∆(G) ≤ χ′(G) ≤ ∆(G) + 1.

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Vizing’s theorem and Kempe equivalence

Theorem (Vizing ’64) For any graph G, ∆(G) ≤ χ′(G) ≤ ∆(G) + 1. Proof through “Kempe changes”.

Marthe Bonamy On Vizing’s edge-colouring question 3/10

Vizing’s theorem and Kempe equivalence

Theorem (Vizing ’64) For any graph G, ∆(G) ≤ χ′(G) ≤ ∆(G) + 1. Proof through “Kempe changes”.

Marthe Bonamy On Vizing’s edge-colouring question 3/10

Vizing’s theorem and Kempe equivalence

Theorem (Vizing ’64) For any graph G, ∆(G) ≤ χ′(G) ≤ ∆(G) + 1. Proof through “Kempe changes”.

Marthe Bonamy On Vizing’s edge-colouring question 3/10

Vizing’s theorem and Kempe equivalence

Theorem (Vizing ’64) For any graph G, ∆(G) ≤ χ′(G) ≤ ∆(G) + 1. Proof through “Kempe changes”.

Marthe Bonamy On Vizing’s edge-colouring question 3/10

Vizing’s theorem and Kempe equivalence

Theorem (Vizing ’64) For any graph G, ∆(G) ≤ χ′(G) ≤ ∆(G) + 1. Proof through “Kempe changes”.

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Vizing’s theorem, revisited

Theorem (Vizing ’64) For any graph G, for any proper edge colouring α of G, there is a proper (∆(G) + 1)-edge colouring β of G such that α and β are Kempe-equivalent.

Marthe Bonamy On Vizing’s edge-colouring question 4/10

Vizing’s theorem, revisited

Theorem (Vizing ’64) For any graph G, for any proper edge colouring α of G, there is a proper (∆(G) + 1)-edge colouring β of G such that α and β are Kempe-equivalent. Conjecture (Vizing ’65) For any graph G, for any proper edge colouring α of G, there is a proper χ′(G)-edge colouring β of G such that α and β are Kempe-equivalent.

Marthe Bonamy On Vizing’s edge-colouring question 4/10

Vizing’s theorem, revisited

Theorem (Vizing ’64) For any graph G, for any proper edge colouring α of G, there is a proper (∆(G) + 1)-edge colouring β of G such that α and β are Kempe-equivalent. Theorem (Misra Gries ’92 (Inspired from the proof)) For any simple graph G = (V , E), a (∆ + 1)-edge-coloring can be found in O(|V | × |E|). Conjecture (Vizing ’65) For any graph G, for any proper edge colouring α of G, there is a proper χ′(G)-edge colouring β of G such that α and β are Kempe-equivalent.

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Vizing’s theorem, revisited

Theorem (Vizing ’64) For any graph G, for any proper edge colouring α of G, there is a proper (∆(G) + 1)-edge colouring β of G such that α and β are Kempe-equivalent. Theorem (Misra Gries ’92 (Inspired from the proof)) For any simple graph G = (V , E), a (∆ + 1)-edge-coloring can be found in O(|V | × |E|). Conjecture (Vizing ’65) For any graph G, for any proper edge colouring α of G, there is a proper χ′(G)-edge colouring β of G such that α and β are Kempe-equivalent. Theorem (Holyer ’81) It is NP-complete to compute χ′.

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Mohar’s conjecture

Conjecture (Vizing ’65) For any graph G, for any proper edge colouring α of G, there is a proper χ′(G)-edge colouring β of G such that α and β are Kempe-equivalent.

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Mohar’s conjecture

Conjecture (Vizing ’65) For any graph G, for any proper edge colouring α of G, there is a proper χ′(G)-edge colouring β of G such that α and β are Kempe-equivalent. Only interesting for χ′(G) = ∆(G).

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Mohar’s conjecture

Conjecture (Vizing ’65) For any graph G, for any proper edge colouring α of G, there is a proper χ′(G)-edge colouring β of G such that α and β are Kempe-equivalent. Only interesting for χ′(G) = ∆(G). Conjecture (Mohar ’06) For any graph G, for any two (∆(G) + 2)-edge colourings α and β

• f G, they are Kempe-equivalent.

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Mohar’s conjecture

Conjecture (Vizing ’65) For any graph G, for any proper edge colouring α of G, there is a proper χ′(G)-edge colouring β of G such that α and β are Kempe-equivalent. Only interesting for χ′(G) = ∆(G). Conjecture (Mohar ’06) For any graph G, for any two (∆(G) + 2)-edge colourings α and β

• f G, they are Kempe-equivalent.

True if χ′(G) = ∆(G).

Marthe Bonamy On Vizing’s edge-colouring question 5/10

Mohar’s conjecture

Conjecture (Vizing ’65) For any graph G, for any proper edge colouring α of G, there is a proper χ′(G)-edge colouring β of G such that α and β are Kempe-equivalent. Only interesting for χ′(G) = ∆(G). Conjecture (Mohar ’06) For any graph G, for any two (∆(G) + 2)-edge colourings α and β

• f G, they are Kempe-equivalent.

True if χ′(G) = ∆(G). (Vizing’s conjecture) ⇒ (Mohar’s conjecture): induction on ∆(G).

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Small Delta

Theorem (McDonald, Mohar, Scheide ’10) Vizing’s conjecture is true for ∆ = 3.

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Small Delta

Theorem (McDonald, Mohar, Scheide ’10) Vizing’s conjecture is true for ∆ = 3. Theorem (Asratian, Casselgren ’16) Vizing’s conjecture is true for ∆ = 4.

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Small Delta

Theorem (McDonald, Mohar, Scheide ’10) Vizing’s conjecture is true for ∆ = 3. Theorem (Asratian, Casselgren ’16) Vizing’s conjecture is true for ∆ = 4. Theorem (B., Defrain, Klimoˇ sov´ a, Lagoutte, Narboni ’20) Vizing’s conjecture is true for triangle-free graphs.

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Small Delta

Theorem (McDonald, Mohar, Scheide ’10) Vizing’s conjecture is true for ∆ = 3. Theorem (Asratian, Casselgren ’16) Vizing’s conjecture is true for ∆ = 4. Theorem (B., Defrain, Klimoˇ sov´ a, Lagoutte, Narboni ’20) Vizing’s conjecture is true for triangle-free graphs. Theorem (B., Defrain, Klimoˇ sov´ a, Lagoutte, Narboni ’20) For any triangle-free graph, all (χ′ + 1)-edge-colourings are Kempe-equivalent.

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General structure

By induction on χ′(G).

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General structure

By induction on χ′(G). It suffices to consider χ′(G)-regular graphs.

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General structure

By induction on χ′(G). It suffices to consider χ′(G)-regular graphs. Consider a target χ′(G)-edge-colouring α, and M one of its color classes (M is a perfect matching).

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General structure

By induction on χ′(G). It suffices to consider χ′(G)-regular graphs. Consider a target χ′(G)-edge-colouring α, and M one of its color classes (M is a perfect matching). Goal: make M monochromatic (say with colour 1).

Marthe Bonamy On Vizing’s edge-colouring question 7/10

General structure

By induction on χ′(G). It suffices to consider χ′(G)-regular graphs. Consider a target χ′(G)-edge-colouring α, and M one of its color classes (M is a perfect matching). Goal: make M monochromatic (say with colour 1). Good (∈ M, coloured 1), bad (∈ M, not coloured 1), ugly (∈ M, coloured 1) edges.

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Fan-like tools

u v 2

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Fan-like tools

u v 2

✁ ❆

3

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Fan-like tools

u v 2

✁ ❆

3

✁ ❆

3

Marthe Bonamy On Vizing’s edge-colouring question 8/10

Fan-like tools

u v 2

✁ ❆

3

✁ ❆

4

Marthe Bonamy On Vizing’s edge-colouring question 8/10

Fan-like tools

u v 2

✁ ❆

3

✁ ❆

4 w 3

✁ ❆

4

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Fan-like tools

u v 2

✁ ❆

3

✁ ❆

4 w 3

✁ ❆

5 x 5

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Fan-like tools

u v 2

✁ ❆

3

✁ ❆

4 w 3

✁ ❆

5 x 5

✁ ❆

3

Marthe Bonamy On Vizing’s edge-colouring question 8/10

Fan-like tools

u v 2

✁ ❆

3

✁ ❆

4 w 3

✁ ❆

5 x 5

✁ ❆

3 − → Dv: vy → vz if vz is coloured with the colour missing at y. uv vw vx

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Fan-like tools

u v 2

✁ ❆

3

✁ ❆

4 w 3

✁ ❆

5 x 5

✁ ❆

3 − → Dv: vy → vz if vz is coloured with the colour missing at y. uv vw vx Xu: sequence of vertices of − → Dv reached from uv. Path:

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Fan-like tools

u v 2

✁ ❆

3

✁ ❆

4 w 3

✁ ❆

5 x 5

✁ ❆

3 − → Dv: vy → vz if vz is coloured with the colour missing at y. uv vw vx Xu: sequence of vertices of − → Dv reached from uv. Path: Cycle:

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Fan-like tools

u v 2

✁ ❆

3

✁ ❆

4 w 3

✁ ❆

5 x 5

✁ ❆

3 − → Dv: vy → vz if vz is coloured with the colour missing at y. uv vw vx Xu: sequence of vertices of − → Dv reached from uv. Path: Cycle: Comet: (sort of)

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Back to the general picture

We can argue the existence of: u v w 1

✁ ❆

1 M

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Back to the general picture

We can argue the existence of: u v w 1

✁ ❆

1 M Then Xv in − → Dw is a cycle.

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Back to the general picture

We can argue the existence of: u v w 1

✁ ❆

1 M Then Xv in − → Dw is a cycle. Xw in − → Dv is also a cycle.

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Back to the general picture

We can argue the existence of: u v w 1

✁ ❆

1 M Then Xv in − → Dw is a cycle. Xw in − → Dv is also a cycle. Two cycles ⇒ (unless there is a triangle vwx with ✁

❆

1 at x).

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Conclusion

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Conclusion Danke sch¨

• n!

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