Gallai-Ramsey Number for K4 Ingo Schiermeyer TU Bergakademie - - PowerPoint PPT Presentation

Chromatic Number

Ingo Schiermeyer

Ingo Schiermeyer

TU Bergakademie Freiberg

(joint work with Colton Magnant and Akira Saito)

JCCA 2018

Gallai-Ramsey Number for K4

JCCA 2018

Chromatic Number

Ingo Schiermeyer

Given two graphs G and H, the Ramsey number R(G,H) is the minimum integer n such that every red/blue-colouring of the edges of the complete graph on n vertices contains either a red copy of G or a blue copy of H.

Gallai-Ramsey Number

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Ingo Schiermeyer

Gallai-Ramsey Number

JCCA 2018

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Ingo Schiermeyer

Gallai-Ramsey Number

165 ) K , R(K 102 48 ) K , R(K 43 18 ) K , R(K 6 ) K , R(K

6 6 5 5 4 4 3 3

     

JCCA 2018

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Ingo Schiermeyer

Gallai-Ramsey Number

Suppose aliens invade the earth and threaten to obliterate it in a year's time unless human beings can find the Ramsey number for red five and blue five. We could marshal the world's best minds and fastest computers, and within a year we could probably calculate the value. If the aliens demanded the Ramsey number for red six and blue six, however, we would have no choice but to launch a preemptive attack. Paul Erdos JCCA 2018

Chromatic Number

Ingo Schiermeyer

Gallai-Ramsey Number

25 ) K , R(K 13 ) K , R(K 9 ) K , R(K

5 4 5 3 4 3

  

JCCA 2018

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Ingo Schiermeyer

Given a graph H, the k-coloured Gallai-Ramsey number is defined to be the minimum integer n such that every k-colouring (using all k colours) of the complete graph on n vertices contains either a rainbow triangle or a monchromatic copy of H.

) H : K ( gr : notation

3 k

Gallai-Ramsey Number

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Ingo Schiermeyer

       

• dd.

is k if 1 5 2 even, is k if 1 5 ) K : K ( gr

1)/2

• (k

k/2 3 3 k

Theorem (Chung and Graham, 1983)

1955) Gleason, and (Greenwood 17 ) K , K , R(K ) K ( R 6 ) K , R(K ) K ( R

3 3 3 3 3 3 3 3 2

   

Gallai-Ramsey Number

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Ingo Schiermeyer

1. k for 4 k 2 ) P : K ( gr 1. k for 4 k ) P : K ( gr 1. k for 3 k ) P : K ( gr 1. k n, for 1 2 n k 2 2

• n

) P : K ( gr 2. k for 4 k ) C : K ( gr

6 3 k 5 3 k 4 3 k n 3 k 4 3 k

                           

Theorem (Faudree, Gould, Jacobson, Magnant, 2010)

Gallai-Ramsey Number

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Ingo Schiermeyer

n. log n ) 3 2 ( ) C : K ( gr 1 n2 n, 3 k ) 1 n ( ) C : K ( gr 1 n 1)k

• (n

1, k and 2 n integers Given . 2 n 3 k 2 2

• n

) P : K ( gr 1 2 n k 2 2

• n

1, k and 3 n integers Given

3 k 1 2n 3 k k 2n 3 k n 3 k

                                          

 

Theorem (Hall, Magnant, Ozeki, Tsugaki, 2014)

Gallai-Ramsey Number

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Ingo Schiermeyer

. 1 k for 1 2 2 ) C : K ( gr . 1 k for 4 k 2 ) C : K ( gr

k 5 3 k 6 3 k

      

Theorem (Fujita and Magnant, 2011)

. 1 k for 1 2 4 ) C : K ( gr . 1 k for 1 2 3 ) C : K ( gr

k 9 3 k k 7 3 k

       

Theorem (Bruce and Song, Bosse and Song, 2017)

. 1 k for 5 k 3 ) C : K ( gr

8 3 k

  

Theorem (Gregory, Magnant and Magnant, 2016)

Gallai-Ramsey Number

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Ingo Schiermeyer

. 1 k for 1 2 5 ) C : K ( gr . 1 k for 1 2 4 ) C : K ( gr

k 11 3 k k 9 3 k

       

Theorem (Bruce and Song, 2017) Theorem (Bosse and Song, 2017)

. 1 k for 1 2 3 ) C : K ( gr

k 7 3 k

   

Gallai-Ramsey Number

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Ingo Schiermeyer

. 1 k and } 4 , 3 { n for 2 n k ) 1 n ( ) P : K ( gr . 1 k for 5 k 4 ) P : K ( gr . 1 k for 5 k 4 ) C : K ( gr

1 2 3 k 10 3 k 10 3 k

            

 n

Theorem (Zhang, Lei, Shi, and Song, 2017)

Gallai-Ramsey Number

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Ingo Schiermeyer

) K , R(K r(p) where

• dd,

is k if 1 1)

• 1)(r(p)
• (p

even, is k if 1 1)

• r(p)

( ) K : K ( gr

p p 1)/2

• (k

k/2 p 3 k

       

Conjecture (Fox, Grinshpun, and Pach, 2015)

Gallai-Ramsey Number

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Ingo Schiermeyer

       

• dd

is k if , 1 17 3 even is k if , 1 17 ) K : K ( gr

1)/2

• (k

k/2 4 3 k

Theorem (Magnant, Saito, and I.S., 2017)

Gallai-Ramsey Number

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Ingo Schiermeyer

Theorem (Gallai, 1967) In any edge-colouring of the complete graph with no rainbow triangle, there exists a partition of the vertices into at least two parts (called a Gallai partition or G-partition for short) such that, there are at most two colours on the edges between the parts, and only one colour on the edges between each pair of parts.

Gallai-Ramsey Number

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Ingo Schiermeyer

17. t 2 have we 18, ) K , R(K Since minimal. be let t and parts, the

• f

number the be Let t blue. and red colours with two partition Gallai a Consider k

• n

induction apply we 3 k For

4 4

     

Proof approach

Gallai-Ramsey Number

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Ingo Schiermeyer

Gallai-Ramsey Number

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Ingo Schiermeyer

Proof in progress – Nihon university, march 2017

Gallai-Ramsey Number

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Proof in progress – Nihon university, march 24, 2017

Gallai-Ramsey Number

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Ingo Schiermeyer

                             even. is s)

• (k

and

• dd,

is s and k, s if 1 5 17 16

• dd,

both are s)

• (k

and s if 1 5 17 8

• dd,

is s and k s if 1 17 3

• dd,

is s)

• (k

and even is s if 1 5 17 2 even, both are s)

• (k

and s if 1 5 17 s) g(k, where s), g(k, ) K ) s

• k

( , sK : K ( gr Then k. s th integer wi an be s and 1, k Let

2 / ) 2

• s
• k

( 1)/2

• (s

2 / ) 1

• s
• k

( 1)/2

• (s

1)/2

• (k

2 / ) 1

• s
• k

( s/2 2 / ) s

• k

( s/2 3 4 3 k

Theorem (Magnant, Saito, and I.S., 2017)

Gallai-Ramsey Number

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The next case: K5 We believe that the conjecture can be proved, although the exact value of R(K5,K5) is not (yet) known. January 16, 2018 – a surprise The complete graph K169 can be coloured with 3 colours such that it contains no monochromatic K5 and no rainbow triangle.

Gallai-Ramsey Number

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Ingo Schiermeyer

The next case: K5 The complete graph K169 can be coloured with 3 colours such that it contains no monochromatic K5 and no rainbow triangle.

Gallai-Ramsey Number

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The next case: K5 Consequences: If R(5,5)=43, then the conjectured value for p=5 and k is odd is false. If 44 <= R(5,5) <= 48, then the conjecture can be true for p=5.

Gallai-Ramsey Number

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The next case: K5

Gallai-Ramsey Number

Suppose aliens invade the earth and threaten to obliterate it in a year's time unless human beings can find the Ramsey number for red five and blue five. We could marshal the world's best minds and fastest computers, and within a year we could probably calculate the value. If the aliens demanded the Ramsey number for red six and blue six, however, we would have no choice but to launch a preemptive attack. Paul Erdos JCCA 2018

Chromatic Number

Ingo Schiermeyer

Dōmo arigatō gozaimasu! Herzlichen Dank!

Thank you very much!

Gallai-Ramsey Number

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The end

Gallai-Ramsey Number