On-line list colouring of graphs

1. On-line list colouring of graphs Xuding Zhu Zhejiang Normal University 2016.8.23 CAM Hongkong

2. A scheduling problem: There are six basketball teams, each needs to compete with all the others. Each team can play one game per day How many days are needed to schedule all the games? Answer: 5 days

3. 1st day

4. 2nd day

5. 3rd day

6. 4th day

7. 5th day

8. This is an edge colouring problem. Each edge is a game. Each day is a colour.

9. A scheduling problem: There are six basketball teams, each needs to compete with all the others. Each team can play one game per day Each team can choose one day off How many days are needed to schedule all the games? Answer: 5 days 7 days are needed 7 days are enough

10. There are 7 colours Edge list colouring Each edge misses at most 2 colours Each edge has 5 permissible colours I do not know any easy proof

11. List colouring conjecture: For any graph G, Haggkvist-Janssen (1997) However, the conjecture remains open for Uwe Schauz (2014)

12. A scheduling problem There are six teams, each needs to compete with all the others. Each team can play one game per day Each team can choose one day off How many days are needed to schedule all the games? Answer: 5 days The choices are made before the scheduling 7 days are enough

13. A scheduling problem There are six teams, each needs to compete with all the others. Each team can play one game per day is allowed not to show up for one day Each team can choose one day off How many days are needed to schedule all the games? On each day, we know which teams haven’t shown up today 7 days are enough but we do not know which teams will not show up tomorrow We need to schedule the games for today

14. On-line list colouring of graphs We start colouring the graph before having the full information of the list

15. is the number of permissible colours for x f-painting game (on-line list colouring game) on G Each vertex v is given f(v) tokens. Each token represents a permissible colour. But we do not know yet what is the colour. Two Players: Lister Painter Colours vertices Reveals the list

16. Lister choose a set of uncoloured vertices, removes one token from each vertex of Painter chooses an independent subset of is the set of vertices which has colour i as a permissible colour. vertices in are coloured by colour i. At round i

17. If at the end of some round, there is an uncolored vertex with no tokens left, then Lister wins. If all vertices are coloured then Painter wins the game.

18. G is f-paintable if Painter has a winning strategy for thef-painting game. G is k-paintable if G is f-paintable for f(x)=k for every x. The paint number of G is the minimum k for which G is k-paintable.

19. On-line list colouring: List colouring: Painter start colouring the graph after having the full information of the list before choice number

20. is not 2-paintable Theorem [Erdos-Rubin-Taylor (1979)] is 2-choosable.

21. is not 2-paintable Lister wins the game

22. Theorem [Erdos-Rubin-Taylor,1979] A connected graph G is 2-choosable if and only if its core is or or However, if p>1, then is not 2-paintable. Theorem [Zhu,2009] A connected graph G is 2-paintable if and only if its core is or or

23. Problems studied Planar graphs and locally planar graphs Chromatic-paintable graphs Complete bipartite graphs Random graphs Partial painting game b-tuple painting game and fractional paint number Defective painting game Sum-painting number of graphs

24. Methods: 1. Derandomize probability arguments 2. Polynomial method 3. Inductive proof 4. Kernel method 5. Probability

25. Complete bipartite graphs Theorem [Erdos,1964] probabilistic proof Theorem[Zhu,2009] If G is bipartite and has n vertices, then

26. A B Probability proof: Each color is assigned to vertices in A or B with probability

27. Assume Lister has given set If Painter colours , double the weight of each vertex in Initially, each vertex x has weight w(x)=1 A B

28. If x has permissible colours, Painter will be able to colour it. A The total weight of uncoloured vertices is not increased. B If a vertex is given a permissible colour but is not coloured by that colour, then its weight doubles. If xhas been given k permissible colours, but remains uncoloured, then

29. Assume Lister has given set If Initially, each vertex x has weight w(x)=1 A B Painter colours , double the weight of each vertex in

30. and Theorem [RadhaKrishnan-Srinivasan,2000] Erdos-Lovasz Conjecture Theorem [Erdos, 1964]

31. and Theorem [RadhaKrishnan-Srinivasan,2000] Erdos-Lovasz Conjecture Theorem [Erdos, 1964] The proof uses a probability argument. The argument CANNOT be derandomized to give a strategy for the painting game.

32. and Theorem [RadhaKrishnan-Srinivasan,2000] Erdos-Lovasz Conjecture Theorem [Erdos, 1964] Theorem [Duray-Gutowski-Kozik,2015] Corollary

33. Some other results proved by derandomizing probabilistic arguments 1: Partial online list colouring

34. Partial painting game Partial f-painting game on G same as the f-painting game, except that Painter’s goal is not to colour all the vertices, but to colour as many vertices as possible.

35. Fact: Conjecture [Albertson]: Conjecture [Zhu, 2009]:

36. Conjecture [Zhu, 2009]: Theorem [Wong-Zhu,2013] Proof: Derandomize a probabilistic argument

37. Some other results proved by derandomizing probabilistic arguments 1: Partial online list colouring 2. Fractional online choice number

38. b-tuple list colouring G is (a,b)-choosable if |L(v)|=a for each vertex v, then there is a b-tuple L-colouring. b-tuple on-line list colouring If each vertex has a tokens, then Painter has a strategy to colour each vertex a set of b colours.

39. Theorem [Alon-Tuza-Voigt, 1997] [Gutowski, 2011] Infimum attained Infimum not attained Probabilistic arguemnt

40. Methods: 1. Derandomize probability arguments 2. Polynomial method

41. paintable = deg(P(G))

42. Some results proved by using polynomial method

43. Haggkvist-Janssen (1997) Uwe Schauz (2014)

44. Methods: 1. Derandomize probability arguments 2. Polynomial method 3. Inductive proof

45. A recursive definition of f-paintable Assume . Then G is f-paintable, if (1) or (2)

46. Upper bounds for ch(G) proved by induction Planar graphs [ Schauz,2009 ] Theorem [Thomassen, 1995] Every planar graph is 5-choosable paintable

47. non-contractible edge-width of G contractible length of shortest non-contractible cycle embedded in a surface Locally planar edge-width is large Theorem [Thomassen, 1993] For any surface , there is a constant , any G embedded in with edge-width > is 5-colourable.

48. non-contractible edge-width of G contractible length of shortest non-contractible cycle embedded in a surface Locally planar edge-width is large Han-Zhu 2015 DeVos-Kawarabayashi-Mohor 2008 Theorem [Thomassen, 1993] For any surface , there is a constant , any G embedded in with edge-width > is 5-colourable. choosable paintable

49. Chromatic-paintable graphs

50. Ohba Conjecture: Graphs G with are chromatic choosable. A graph G is chromatic choosable if paintable Conjecture: Line graphs are chromatic choosable. paintable Conjecture: Claw-free graphs are chromatic choosable. paintable Conjecture: Total graphs are chromatic choosable. paintable [Kim-Park,2013] Conjecture: Graph squares are chromatic choosable. Theorem [Noel-Reed-Wu,2013] paintable