On-line list colouring of graphs

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

2. Suppose G is a graph A list assignment L assigns to each vertex x a set L(x) of permissible colours. An L-colouring h of G assigns to each vertex x a colour such that for every edge xy choice number

3. Given a colour i, is the set of vertices having i as a permissible colour. Given a vertex x, L(x) tells us which colours are permissible. Alternately, given a colour i, one can ask which vertices have i as a permissible colour. A list assignment L can be given as An L-colouring is afamily of independent subsets and

4. is the number of permissible colours for x At round i, Alice choose a set of uncoloured vertices. is the set of vertices which has colour i as a permissible colour. Bob chooses an independent subset of and colour vertices in by colour i. on-line f-list colouring game on G played by Alice and Bob Alice wins the game if there is a vertex x, which has been given f(x) permissible colours and remains uncoloured. Otherwise, eventually all vertices are coloured and Bob wins the game.

5. The on-line choice number of G is the minimum k for which G is on-line k-choosable. G is on-line f-choosable if Bob has a winning strategy for the on-line f-list colouring game. G is on-line k-choosable if G is on-line f-choosable for f(x)=k for every x.

6. is not on-line 2-choosable Theorem [Erdos-Rubin-Taylor (1979)] is 2-choosable.

7. is not on-line 2-choosable Alice wins the game

8. Question: Can the difference be arbitrarily large ? Question: Can the ratio be arbitrarily large ?

9. Theorem [Schauz,2009] For planar G, Theorem [Chung-Z,2011] For planar G, triangle free + no 4-cycle adjacent to a 4-cycle or a 5-cycle, Most upper bounds for choice number are also upper bounds for on-line choice number. Currently used method in proving upper bounds for choice number Induction Some works for on-line choice number, Kernel method Combinatorial Nullstellensatz Theorem [Schauz,2009] Upper bounds for ch(G) proved by Combinatorial nullstellensatz works for on-line choice number

10. Theorem [Schauz,2009] If G has an orientation D with then G is on-line The proof is by induction (no polynomial is involved). Theorem [Schauz,2009] Upper bounds for ch(G) proved by Combinatorial nullstellensatz works for on-line choice number

11. Probabilistic method Does not work for on-line choice number Theorem [Alon, 1992] The proof is by probabilistic method Theorem[Z,2009] Proof: If G is bipartite and has n vertices, then

12. Assume Alice has given set If Bob colours , double the weight of each vertex in If a vertex x has permissible colours, Bob will be able to colour it. Initially, each vertex x has weight w(x)=1 A B The total weight of uncoloured vertices is not increased. If a vertex is given a permissible colour but is not coloured by that colour, then it weight doubles. If a vertex x has given k permissible colours, but remains uncoloured, then

13. Conjecture [Ohba] Graphs G with are chromatic choosable. Conjecture: For any G, for any k > 1, is chromatic choosable. A graph G is chromatic choosable if Conjecture: Line graphs are chromatic choosable. Conjecture: Claw-free graphs are chromatic choosable. Conjecture: Total graphs are chromatic choosable. Theorem [Noel-Reed-Wu]

14. On-line version of Ohba Conjecture: Graphs G with are on-line chromatic choosable. NOT TRUE 3 3 33|33|333 3 3 3 3 3

15. On-line version of Ohba Conjecture: Graphs G with are on-line chromatic choosable. 3 3 33|33|333 3 3 Alice’s move 3 3 3

16. On-line version of Ohba Conjecture: Graphs G with are on-line chromatic choosable. 3|23|233 3 3 33|33|333 3 3 23|23|33 Alice’s move 3 3 3 Bob’s (2 possible) moves

17. On-line version of Ohba Conjecture: Graphs G with are on-line chromatic choosable. 13|222 3|23|233 3 3 2|3|222 33|33|333 3 3 23|23|33 Alice’s move 3 3 3 Bob’s (2 possible) moves

18. 3|111 On-line version of Ohba Conjecture: Graphs G with are on-line chromatic choosable. 13|222 3|23|233 3 3 2|3|222 33|33|333 3 3 23|23|33 Alice’s move 3 3 3 Bob’s (2 possible) moves

19. 3|111 On-line version of Ohba Conjecture: Graphs G with are on-line chromatic choosable. 13|222 3|23|233 3 3 2|3|222 33|33|333 3 3 2|112 23|23|33 Alice’s move 3 3 3 Bob’s (2 possible) moves

20. 3|111 On-line version of Ohba Conjecture: Graphs G with are on-line chromatic choosable. 13|222 3|23|233 3 3 2|3|222 33|33|333 3 3 2|112 23|23|33 Alice’s move 3 3 3 3|13|22 2|3|11

21. Theorem [Kim-Kwon-Liu-Z,2012] For n > 1, is not on-line -choosable

22. On-line version of Ohba Conjecture: Graphs G with are on-line chromatic choosable.

23. If for some k, have a common permissible colour and Theorem [Erdos-Rubin-Taylor (1979)] is chromatic choosable. Proof Assume each vertex is given n permissible colours. then colour them by a common colour use induction to colour the rest.

24. Assume no partite set has a common permissible colour Build a bipartite graph V The proof does not work for on-line list colouring C colours By Hall’s theorem, there is a matching that covers all the vertices of V

25. On-line version of Ohba Conjecture: Graphs G with are on-line chromatic choosable. Question: is on-line n-choosable ?

26. Question: Theorem [Huang-Wong-Z,2010] is on-line n-choosable. Proof Combinatorial Nullstellensatz An explicit winning strategy for Bob ( Kim-Kwon-Liu-Z, 2012)

27. Theorem [Kim-Kwon-Liu-Z, 2012] G: complete multipartite graph with partite sets satisfying the following i.e., G is on-line f-choosable. Then (G,f) is feasible, is on-line n-choosable.

28. Alice’s choice Bob’s choice After this round, G changed and f changed. Need to prove: new (G,f) still satisfies the condition

29. Alice’s choice

30. Alice’s choice Bob colours v

31. Alice’s choice

32. Alice’s choice Bob colours v

33. Theorem [Kim-Kwon-Liu-Z, 2012] G: complete multipartite graph with partite sets satisfying the following i.e., G is on-line f-choosable. Then (G,f) is feasible, is on-line n-choosable.

34. Theorem [Kozik-Micek-Z, 2012] G: complete multipartite graph with partite sets satisfying the following Then (G,f) is feasible.

35. On-line version of Ohba Conjecture: Graphs G with are on-line chromatic choosable. Conjecture holds for graphs with independence number 3

36. Open Problems Can the difference ch^{OL}-ch be arbitrarily big? On-line Ohba conjecture true?

37. Nine Dragon Tree Thank you