1. On polynomial chi-binding functions
Ingo Schiermeyer
TU Bergakademie Freiberg
BIRS 2016 1

2. Chromatic Number Theorem (Erdős, 1959)
Coloring, sparseness, and girth, 2015
Alon, Kostochka, Reiniger, West, and Zhu

3. Chromatic Number
Mycielski construction

4. Chromatic Number Strong Perfect Graph Conjecture (Berge, 1960)
A graph G is perfect if and only if it does not contain an odd hole nor an odd antihole of length at least five.
The SPGC became the
Strong Perfect Graph Theorem in 2002 by Chudnovsky, Robertson, Seymour, and Thomas.

5. Chromatic Number András Gyárfás (Budapest, 1985)
Problems from the world surrounding perfect graphs

6. Chromatic Number Conjecture (Gyárfás)
Let T be any tree (or forest). Then there is a function
( -binding function) such that every T-free graph G satisfies

7. Chromatic Number
This paper of (Gyárfás) contains several other challenging conjectures.
Project „Induced subgraphs of graphs with large chromatic number“ by Chudnovsky, Scott, and Seymour.
Theorem (CSS 2014)
Every pentagonal graph is colourable.

8. Chromatic Number Theorem (Gyárfás, 1987)
There is a chi-binding function for
Stars
Paths
Brooms
Trees with radius 2 (Kierstead and Penrice)
Special trees with radius 3 (Kierstead and Zhu)
pK2 (Wagon 1980)

9. Chromatic Number Problem 2.7 (Gyárfás, 1987)

10. Chromatic Number Problem 2.7 (Gyárfás, 1987)
If there is a polynomial binding function for P5-free graphs,
then it would imply the
Erdös-Hajnal Conjecture for P5-free graphs, which is open.
Problem 2.7 (Gyárfás, 1987)
Theorem (Gyárfás,1987)
There is no linear binding function for P5-free graphs.

11. Chromatic Number
Erdös-Hajnal Conjecture

12. Induced subgraphs
Dart
Gem
House
Bull
Kite

13. Induced subgraphs
K_2,3
House

14. Chromatic Number Problem 2.16 (Gyárfás, 1987) Theorem (Wagon, 1980)
Theorem (Chung, 1980)
There is no linear binding function.
Theorem (BRSV, 2016)

15. Chromatic Number Problem 2.16 (Gyárfás, 1987) Theorem (Wagon, 1980)
Work in progress

16. Chromatic Number
? ?

17. Chromatic Number
? ?

18. Chromatic Number
? ?

19. Chromatic Number
? ?

20. Chromatic Number
? ?

21. Chromatic Number

22. Chromatic Number
Theorem (Fouquet et al. 1995)
Theorem (IS 2014)
Gem+

23. Induced subgraphs
Gem+
TwinGem

24. Chromatic Number
Problem

25. Chromatic Number P5 ? ? ? ? ? P5, gem+ P5, cricket P5, dart P5, gem
? ? ?
? ?
P5, gem+
P5, cricket
P5, dart
P5, gem
P5,hammer
P5, windmill
P5,claw
P5, paw
P5, diamond
P5, house
P5, K2,t

26. Chromatic Number Theorem (Gyárfás, 1987)
Theorem (Esperet, Lemoine, Maffray, Morel, 2013)

27. Chromatic Number
Theorem (Gyárfás, 1987)

28. Chromatic Number Dōmo arigatō gozaimasu! Herzlichen Dank!
Mille grazie!
Thank you very much!

29. The end

30. Chromatic Number
Theorem (BDS 2015)

31. Chromatic Number
Sketch of proof:

32. Induced subgraphs
Butterfly -- Windmill (3,2)
CTW 2016

33. Chromatic Number
CTW 2016

34. SEG 2016

35. Chromatic Number Problems:
We are still confused, but on a higher level (Feynmann)

36. Induced subgraphs
Dart
Gem
House
Bull
Kite

37. Chromatic Number
Theorem (Fouquet et al. 1995)
Theorem (IS 2014)
Gem+

38. Induced subgraphs
Gem+
TwinGem

39. Chromatic Number
Problem

40. Chromatic Number András Gyárfás (Budapest, 1987)
Problems from the world surrounding perfect graphs
AGTAC 2015

41. Chromatic Number Conjecture (Gyárfás)
Let T be any tree (or forest). Then there is a function
( -binding function) such that every T-free graph G satisfies
SEG 2015

42. Chromatic Number
Mycielski construction
SEG 2015

43. Chromatic Number Theorem (Gyárfás, 1987)
Theorem (Esperet, Lemoine, Maffray, Morel, 2013)
SEG 2015

44. Chromatic Number Problem (Gyárfás,1987)
If there is a polynomial binding function for P5-free graphs,
then it would imply the
Erdös-Hajnal Conjecture for P5-free graphs, which is open.
Theorem (Gyárfás,1987, IS 2014)
There is no linear binding function for P5-free graphs.
SEG 2015

45. Chromatic Number Theorem (Fouquet et al. 1995) Theorem (IS 2014)
SEG 2015

46. Chromatic Number
Theorem (Brooks, 1941)
SEG 2015

47. Chromatic Number Conjecture (Borodin and Kostochka, 1977)
Proved for claw-free graphs by Cranston and Rabern 2012.
Theorem (Rabern, 2014)
SEG 2015

48. Chromatic Number
Conjecture (Reed, 1998)
SEG 2015

49. Chromatic Number
Reed‘s conjecture holds for
SEG 2015

50. Chromatic Number
Reed‘s conjecture holds for
SEG 2015

51. Induced subgraphs
Dart
Gem
House
Bull
Kite
SEG 2015

52. Chromatic Number
Theorem (IS 2014)
SEG 2015

53. Chromatic Number
Theorem (IS 2014)
SEG 2015

55. Chromatic Number
Erdös, P., Simonovits, M., Sos, V.T.: Anti-Ramsey theorems.
In: Infinite and Finite Sets (Proc.Colloq. Keszthely, Hungary 1973) edited by Hajnal, A., Rado, R., Sos, V.T. vol. II, pp
Colloq. Math. Soc. Janos Bolyai 10. Amsterdam: North-Holland 1975
MCW XX

56. Chromatic Number
f(G,H): Maximum number of colours in an edge colouring of G with no totally multicoloured copy of H.
If G is coloured with at least f(G,H) + 1 = rb(G,H) colours, then there is a totally multicoloured copy or a Rainbow copy of H.
MCW XX

57. Chromatic Number
Let Then
MCW XX

58. Chromatic Number Turan numbers Extremal graph theory Rainbow numbers
MCW XX

59. Chromatic Number
Theorem (Turán, 1941)
MCW XX

60. Chromatic Number Theorem (Erdös, Simonovits and Sós, 1975)
Theorem (Montellano-Ballesteros and Neumann-Lara, Schiermeyer, 2002)
MCW XX

61. Rainbow Matchings Theorem (Erdös & Gallai, 1959)
Theorem (Schiermeyer, 2004)
MCW XX

62. Rainbow Matchings
MCW XX

63. Rainbow Cycles Conjecture (Erdös, Simonovits and Sós, 1975)
with n = t(k-1) + r and
This conjecture was proved for k=4 by Alon (1983), for k = 5, 6 and 7
by Jiang, West and Schiermeyer (2001, 2003), and for all k by
Montellano-Ballesteros and Neumann-Lara (2005).
MCW XX

64. Rainbow Cycles
MCW XX

65. Rainbow Cycles Theorem (Montellano-Ballesteros, Neumann-Lara, 2005)
with n = t(k-1) + r and
MCW XX

66. Rainbow Cycles Theorem (Gorgol, 2008)
where denotes a cycle with a pendant edge.
Theorem (Gorgol, 2008)
where denotes a cycle with two pendant edges.
MCW XX

67. Rainbow bulls and diamonds
Bull B
Diamond D
MCW XX

68. Rainbow numbers for certain graphs
Theorem (Gorgol and Łazuka, 2006)
Theorem (Neupauerova, Sotak, IS 2012)
MCW XX

69. Rainbow numbers for diamonds
Theorem (IS, 2012)
Theorem (IS, 2012)
Let H be a graph of order containing a cycle
If H has cyclomatic number , then
has no upper bound which is linear in n.
MCW XX

70. Rainbow numbers for diamonds
Theorem (Garnick, Kwong, and Lazebnik, 1993)
Conjecture (Erdös, 1975)
Theorem (Garnick, Kwong, and Lazebnik, 1993)
MCW XX

71. Rainbow numbers for diamonds
Theorem (IS, 2012)
MCW XX

72. Rainbow numbers for diamonds
Theorem (IS, 2012)
MCW XX

73. Rainbow numbers for diamonds
MCW XX

74. Rainbow numbers for diamonds
MCW XX

75. Rainbow Colourings n D H
K(2,3) 4 5 7 8 6 9 10 11 12 14 17
MCW XX

76. GRAPHS 2008