Xuding Zhu Zhejiang Normal University

1. Circular colouring of graphs Xuding Zhu Zhejiang Normal University

2. A distributed computation problem V: a set of computers D: a set of data files

3. a b e c d If x ~ y, then x and y cannot operate at the same time. If x ~ y, then x and y must alternate their turns in operation.

4. Schedule the operating time of the computers efficiently. Efficiency: proportion of computers operating on the average. The computer time is discrete: time 0, 1, 2, …

5. 1: colouring solution Colour the vertices of G with k= colours.

6. a The efficiency is 1/3 0 2 e b 1 1 0 d c Colour the graph with 3 colours 5 4 At time 0 1 2 3 e e a c a c b d b d Operate machines

7. The efficiency is In general, at time t, those vertices x with f(x)=t mod (k) operate.

8. Computer scientists solution

10. The efficiency of this scheduling is Better than the colouring solution

11. If initially the keys are assigned as above, then no computer can operate.

12. Once an orientation is given, then the scheduling is determined. The problem of calculating the efficiency (for a given orientation) is equivalent to a well studied problem in computer science: the minimum cycle mean problem

13. Theorem [Barbosa et al. 1989] There is an initial assignment of keys such that the scheduling derived from such an assignment is optimal.

14. Circular colouring of graphs

15. G=(V,E): a graph 0 an integer 1 1 An k-colouring of G is 2 0 such that A 3-colouring of

16. The chromatic number of G is

17. G=(V,E): a graph 0 a real number 1 an integer 1.5 A (circular) k-colouring of G is r-colouring of G is An 2 0.5 A 2.5-coloring such that

18. The circular chromatic number of G is { r: G has a circular r-colouring } min inf

19. f is k-colouring of G f is a circular k-colouring of G Therefore for any graph G,

20. 0=r 0 r 1 4 2 3 |f(x)-f(y)|_r ≥ 1 x~y p p’ The distance between p, p’ in the circle is f is a circular r-colouring if

21. Circular coloring method Let r= Let f be a circular r-coloring of G x operates at time k iff for some integer m

22. r=4.4 0 4 1 3 2 4 0 5 3 1 2

23. The efficiency of this scheduling is Theorem [Yeh-Zhu] The optimal scheduling has efficiency

24. is a refinement of : the real chromatic number is an approximation of

25. Vince, 1988. • star chromatic number More than 300 papers published.

26. “The theory of circularcolorings of graphs has become an important branch of chromatic graph theory with many exciting results and new techniques.” It stimulates challenging problems, leads to better understanding of graph structure in terms of colouring parameters.

27. Interesting questions for are usually also interesting for There are also questions that are not interesting for , but interesting for For the study of , one may need to sharpen the tools used in the study of

28. Questions interesting for both and

29. For which rational , there is a graph G with Answer [Vince 1988]:

30. For which rational , there is a graph G with Answer (Erdos classical result): all positive integers.

31. For which rational , there is a graph G with Answer [Zhu, 1996]:

32. For which rational , there is a graph G with Four Colour Theorem

33. For which rational , there is a graph G with Four Colour Theorem implies Answer [Moser, Zhu, 1997]:

34. For which rational , there is a graph G with Hadwiger Conjecture

35. For which rational , there is a graph G with Hadwiger Conjecture implies Answer [Liaw-Pan-Zhu, 2003]: Answer [Hell- Zhu, 2000]:

36. For which rational , there is a graph G with Hadwiger Conjecture implies Answer [Liaw-Pan-Zhu, 2003]: Answer [Hell- Zhu, 2000]: [Pan- Zhu, 2004]:

37. For which rational , there is a graph G with Trivially: all positive integers

38. For which rational , there is a graph G with We know very little

41. What happens in the interval [3,4]? Theorem [Afshani-Ghandehari-Ghandehari-Hatami-Tusserkani-Zhu,2005]

42. What happens in the interval [3,4]? Maybe it will look like the interval [2,3]: Gaps everywhere ? NO! Theorem [Afshani-Ghandehari-Ghandehari-Hatami-Tusserkani-Zhu,2005]

43. What happens in the interval [3,4]? Theorem [Lukot’ka-Mazak,2010]

44. Theorem [Lukot’ka-Mazak,2010]

45. Theorem [Lin-Wong-Zhu,2013] Theorem [Lin-Wong-Zhu,2013] Theorem [Lukot’ka-Mazak,2010]

46. For the study of , one may need to sharpen the tools used in the study of

47. A powerful tool in the study of list colouring graphs is Combinatorial Nullstellensatz Give G an arbitrary orientation. Find a proper colouring= find a nonzero assignment to a polynomial

49. What is the polynomial for circular colouring? Give G an arbitrary orientation.

50. 0 3 2 4 1