4-colorability of P6-free graphs Ingo Schiermeyer TU Bergakademie - - PowerPoint PPT Presentation

Chromatic Number

Ingo Schiermeyer

Ingo Schiermeyer

TU Bergakademie Freiberg AGH Cracow

4-colorability of P6-free graphs

Joint work with Christoph Brause, Prěmysl Holub, Zdeněk Ryjáček, Petr Vrána, Rastislav Krivoš-Belluš

Chromatic Number

Ingo Schiermeyer

Chromatic Number

Theorem (Randerath, IS, and Tewes, 2002)

graphs. free

• P

for time polynomial in solved be can problem ty colorabili

• 3

The

5

Theorem (Randerath and IS, 2004)

graphs. free

• P

for time polynomial in solved be can problem ty colorabili

• 3

The

6

Chromatic Number

Ingo Schiermeyer

Chromatic Number

graphs. free

• P

for time polynomial in solved be can problem ty colorabili

• 3

The

7

Theorem (Chudnovsky et al., 2014)

Chromatic Number

Ingo Schiermeyer

Chromatic Number

Theorem (Hoang, Kaminski, Lozin, Sawada, and Shu, 2010)

graphs. free

• P

for time polynomial in decided can problem ty colorabili

• k

The

5

Theorem (Huang, 2013)

7. t all for graphs free

• P
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class for the complete

• NP

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• 4

The 6. t 5, k all for graphs free

• P
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class for the complete

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• k

The

t t

  

Chromatic Number

Ingo Schiermeyer

Chromatic Number

Theorem (Kaminski and Lozin, 2007)

complete.

• NP

is g most at length

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cycles no with graphs for problem ty colorabili

• k

the 3, g k, every For 

Theorem (Kaminski and Lozin, 2007)

complete.

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remains graphs free

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for problem ty colorabili

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the k, every For cycle. a containing graph a be H Let

Chromatic Number

Ingo Schiermeyer

Chromatic Number

Theorem (Kaminski and Lozin, 2007)

complete.

• NP

is graphs free

• claw

for problem ty colorabili

• k

the 3, k every For 

Theorem (Kaminski and Lozin, 2007)

complete.

• NP

remains graphs free

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for problem ty colorabili

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the k, every For claw. a containing graph a be H Let

Chromatic Number

Ingo Schiermeyer

Chromatic Number

Theorem (Kaminski and Lozin, 2007)

path. a is H

• f

component every then time, polynomial in solved be can graphs free

• H

for problem ty colorabili

• k

the If graph. a H and integer, an be 3 k Let 

Chromatic Number

Ingo Schiermeyer

Chromatic Number

Theorem (Hoang, Kaminski, Lozin, Sawada, and Shu, 2010)

graphs. free

• P

for time polynomial in decided can problem ty colorabili

• k

The

5

Theorem (Huang, 2013)

7. t all for graphs free

• P
• f

class for the complete

• NP

is problem ty colorabili

• 4

The 6. t 5, k all for graphs free

• P
• f

class for the complete

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is problem ty colorabili

• k

The

t t

  

Chromatic Number

Ingo Schiermeyer

Chromatic Number

Conjecture (Huang, 2013)

graphs. free

• P

for time polynomial in decided be can problem ty colorabili

• 4

The

6

Chromatic Number

Ingo Schiermeyer

Induced subgraphs

Paw Banner

Chromatic Number

Ingo Schiermeyer

Chromatic Number

Theorem (Randerath, IS, and Tewes, 2002)

graphs. such coloring

• 4

for algorithm time polynomial a is there and colorable

• 4

is graph free

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K , (P Every

3 6

Theorem (Huang, 2013)

graphs. free

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, (P

• f

class for the time polynomial in solved be can problem ty colorabili

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The

6

Chromatic Number

Ingo Schiermeyer

Chromatic Number

Theorem (Lozin and Rautenbach, 2003) Theorem (Huang, 2013)

graphs. free

• banner)

, (P

• f

class for the time polynomial in solved be can problem ty colorabili

• 4

The

6

3. r any for graphs free

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K , (P

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class for the time polynomial in solved be can problem ty colorabili

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The

r 1, 6



Chromatic Number

Ingo Schiermeyer

Chromatic Number

Theorem (Chudnovsky, Maceli, Stacho, and Zhong, 2014)

graphs. free

• )

C , (P

• f

class for the time polynomial in solved be can problem ty colorabili

• 4

The

5 6

Chromatic Number

Ingo Schiermeyer

Induced subgraphs

Bull Kite Chair Paw Z2 Banner

Chromatic Number

Ingo Schiermeyer

Chromatic Number

Theorem (BHKRSV, 2015) Theorem (BHKRSV, 2015)

graphs. free

• chair)

, (P

• f

class for the time polynomial in solved be can problem ty colorabili

• 4

The

6

graphs. free

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bull, , (P

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• 4

The graphs. free

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Z bull, , (P

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class for the time polynomial in solved be can problem ty colorabili

• 4

The

6 2 6

Chromatic Number

Ingo Schiermeyer

Rainbow Colourings

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