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

Gallai-Ramsey Number for K4 Ingo Schiermeyer TU Bergakademie Freiberg (joint work with Colton Magnant and Akira Saito ) JCCA 2018 JCCA 2018 Chromatic Number Ingo Schiermeyer

Gallai-Ramsey Number 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. JCCA 2018 Chromatic Number Ingo Schiermeyer

Gallai-Ramsey Number JCCA 2018 Chromatic Number Ingo Schiermeyer

Gallai-Ramsey Number  R(K , K ) 6 3 3  R(K , K ) 18 4 4   43 R(K , K ) 48 5 5   102 R(K , K ) 165 6 6 JCCA 2018 Chromatic Number 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  R(K , K ) 9 3 4  R(K , K ) 13 3 5  R(K , K ) 25 4 5 JCCA 2018 Chromatic Number Ingo Schiermeyer

Gallai-Ramsey Number 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. notation : gr ( K : H ) k 3 JCCA 2018 Chromatic Number Ingo Schiermeyer

Gallai-Ramsey Number Theorem (Chung and Graham, 1983)   k/2 5 1 if k is even,   gr ( K : K )  k 3 3   (k - 1)/2  2 5 1 if k is odd.   R ( K ) R(K , K ) 6 2 3 3 3   R ( K ) R(K , K , K ) 17 (Greenwood and Gleason, 1955) 3 3 3 3 3 JCCA 2018 Chromatic Number Ingo Schiermeyer

Gallai-Ramsey Number Theorem (Faudree, Gould, Jacobson, Magnant, 2010)    gr ( K : C ) k 4 for k 2. k 3 4     n - 2 n     gr ( K : P ) k 1 for n, k 1.     k 3 n     2 2    gr ( K : P ) k 3 for k 1. k 3 4    gr ( K : P ) k 4 for k 1. k 3 5    gr ( K : P ) 2 k 4 for k 1. k 3 6 JCCA 2018 Chromatic Number Ingo Schiermeyer

Gallai-Ramsey Number Theorem (Hall, Magnant, Ozeki, Tsugaki, 2014)   Given integers n 3 and k 1,         n - 2 n n - 2 n      k 1 gr ( K : P ) k 3 .         k 3 n         2 2 2 2   Given integers n 2 and k 1,       (n - 1)k n 1 gr ( K : C ) ( n 1 ) k 3 n, k 3 2n      k k 3 n2 1 gr ( K : C ) ( 2 3 ) n log n.  k 3 2n 1 JCCA 2018 Chromatic Number Ingo Schiermeyer

Gallai-Ramsey Number Theorem (Fujita and Magnant, 2011)    gr ( K : C ) 2 k 4 for k 1 . k 3 6     k gr ( K : C ) 2 2 1 for k 1 . k 3 5 Theorem (Gregory, Magnant and Magnant, 2016)    gr ( K : C ) 3 k 5 for k 1 . k 3 8 Theorem (Bruce and Song, Bosse and Song, 2017)     k gr ( K : C ) 3 2 1 for k 1 . k 3 7     k gr ( K : C ) 4 2 1 for k 1 . k 3 9 JCCA 2018 Chromatic Number Ingo Schiermeyer

Gallai-Ramsey Number Theorem (Bruce and Song, 2017)     k gr ( K : C ) 3 2 1 for k 1 . k 3 7 Theorem (Bosse and Song, 2017)     k gr ( K : C ) 4 2 1 for k 1 . k 3 9     k gr ( K : C ) 5 2 1 for k 1 . k 3 11 JCCA 2018 Chromatic Number Ingo Schiermeyer

Gallai-Ramsey Number Theorem (Zhang, Lei, Shi, and Song, 2017)    gr ( K : C ) 4 k 5 for k 1 . k 3 10    gr ( K : P ) 4 k 5 for k 1 . k 3 10        gr ( K : P ) ( n 1 ) k n 2 for n { 3 , 4 } and k 1 .  k 3 2 1 n JCCA 2018 Chromatic Number Ingo Schiermeyer

Gallai-Ramsey Number Conjecture (Fox, Grinshpun, and Pach, 2015)   k/2 ( r(p) - 1) 1 if k is even,   gr ( K : K )  k 3 p  (k - 1)/2  (p - 1)(r(p) - 1) 1 if k is odd,  where r(p) R(K , K ) p p JCCA 2018 Chromatic Number Ingo Schiermeyer

Gallai-Ramsey Number Theorem (Magnant, Saito, and I.S., 2017)   k/2 17 1 , if k is even   gr ( K : K )  k 3 4   (k - 1)/2  3 17 1 , if k is odd JCCA 2018 Chromatic Number Ingo Schiermeyer

Gallai-Ramsey Number 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. JCCA 2018 Chromatic Number Ingo Schiermeyer

Gallai-Ramsey Number Proof approach   For k 3 we apply induction on k  Consider a Gallai partition with two colours red and blue. Let t be the number of the parts, and let t be minimal. Since    R(K , K ) 18, we have 2 t 17. 4 4 JCCA 2018 Chromatic Number Ingo Schiermeyer

Gallai-Ramsey Number JCCA 2018 Chromatic Number Ingo Schiermeyer

Gallai-Ramsey Number Proof in progress – Nihon university, march 2017 JCCA 2018 Chromatic Number Ingo Schiermeyer

Gallai-Ramsey Number Proof in progress – Nihon university, march 24, 2017 JCCA 2018 Chromatic Number Ingo Schiermeyer

Gallai-Ramsey Number Theorem (Magnant, Saito, and I.S., 2017)    Let k 1, and s be an integer wi th 0 s k. Then  gr ( K : sK , ( k - s ) K ) g(k, s), where k 3 4 3    s/2 ( k - s ) / 2 17 5 1 if s and (k - s) are both even,      s/2 ( k - s - 1 ) / 2 2 17 5 1 if s is even and (k - s) is odd,      (k - 1)/2  g(k, s) 3 17 1 if s k and s is odd,     (s - 1)/2 ( k - s - 1 ) / 2 8 17 5 1 if s and (k - s) are both odd,       (s - 1)/2 ( k - s - 2 ) / 2 16 17 5 1 if s k, and s is odd, and (k - s) is even.  JCCA 2018 Chromatic Number Ingo Schiermeyer

Gallai-Ramsey Number 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. JCCA 2018 Chromatic Number Ingo Schiermeyer

Gallai-Ramsey Number 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. JCCA 2018 Chromatic Number Ingo Schiermeyer

Gallai-Ramsey Number 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. JCCA 2018 Chromatic Number Ingo Schiermeyer

Gallai-Ramsey Number The next case: K5 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 Dōmo arigatō gozaimasu! Herzlichen Dank! Cảm ơn bạn đã quan tâm Thank you very much! JCCA 2018 Chromatic Number Ingo Schiermeyer

Gallai-Ramsey Number The end Chromatic Number Ingo Schiermeyer