1. Online Vertex-Coloring Games in Random Graphs Reto Spöhel (joint work with Martin Marciniszyn; appeared at SODA ’07)

2. Introduction Ramsey theory: when are the edges/vertices of a graph colorable with r colors without creating a monochromatic copy of some fixed graph F ? We call such colorings valid (with respect to F ).

3. Introduction Ramsey theory: when are the edges/vertices of a graph colorable with r colors without creating a monochromatic copy of some fixed graph F ? For random graphs: solved in full generality by [ Ł uczak, Ruci ń ski, Voigt (1992)] (vertex colorings) [Rödl, Ruci ń ski (1995)] (edge colorings)

4. Introduction Ramsey theory: when are the edges/vertices of a graph colorable with r colors without creating a monochromatic copy of some fixed graph F ? For random graphs: solved in full generality by [ Ł uczak, Ruci ń ski, Voigt (1992)] (vertex colorings) [Rödl, Ruci ń ski (1995)] (edge colorings) Throughout this talk: G n, p denotes the random graph on n vertices obtained by including each possible edge with probability p = p ( n ) independently. We consider the vertex-coloring case

5. Introduction [ Ł uczak, Ruci ń ski, Voigt (1992)]: For any fixed graph F and any fixed number of colors r ¸ 2, there are explicit threshold functions p 0 ( F, r, n ) such that In fact, p 0 ( F, r, n ) = p 0 ( F, n ), i.e., the threshold does not depend on the number of colors r e.g., p 0 ( K 3, 2, n )= p 0 ( K 3, 1000, n )= n - 2 / 3 We transfer this result into an online setting, where the vertices of G n, p have to be colored one by one before seeing the entire graph.

6. The online vertex-coloring game Rules: random graph G n, p, initially hidden vertices are revealed one by one along with induced edges vertices have to be instantly (‚online‘) colored with one of r R 2 available colors. game ends when monochromatic copy of some fixed forbidden graph F appears Question: How dense can the underlying random graph be such that Painter can color all vertices a.a.s.?

7. Example F = K 3, r = 2

8. Main result (simplified) Theorem (Marciniszyn, S.; SODA ’07) Let F be a clique or a cycle of arbitrary size. Then the threshold for the online vertex-coloring game with respect to F and with r R 2 available colors is i.e.,

9. Bounds from ‚offline‘ graph properties G n, p contains no copy of F  Painter wins with any strategy G n, p allows no r -vertex-coloring avoiding F  Painter loses with any strategy  the thresholds of these two ‚offline‘ graph properties bound p 0 ( n ) from below and above.

10. Appearance of small subgraphs Theorem (Bollobás, 1981) Let F be a non-empty graph. The threshold for the graph property ‚ G n, p contains a copy of F ‘ is where

11. Appearance of small subgraphs m ( F ) is half of the average degree of the densest subgraph of F. For ‚nice‘ graphs – e.g. for cliques or cycles – we have (such graphs are called balanced)

12. Vertex-colorings of random graphs Theorem ( Ł uczak, Ruci ń ski, Voigt, 1992) Let F be a graph and let r R 2. The threshold for the graph property ‚every r -vertex-coloring of G n, p contains a monochromatic copy of F ‘ is where

13. Vertex-colorings of random graphs For ‚nice‘ graphs – e.g. for cliques or cycles – we have (such graphs are called 1-balanced) If F is 1-balanced, is also the threshold for the property ‚There are more than n copies of F in G n, p ‘ Intuition: For p [ p 0, the copies of F overlap in vertices, and coloring G n, p becomes difficult.

14. For arbitrary F and r we thus have Theorem Let F be a clique or a cycle of arbitrary size. Then the threshold for the online vertex-coloring game with respect to F and with r R 1 available colors is r = 1  Small Subgraphs r    exponent tends to exponent for offline case Main result revisited

15. Lower bound ( r = 2 ) Let p ( n ) / p 0 ( F, 2, n ) be given. We need to show: There is a strategy which allows Painter to color all vertices of G n, p a.a.s.

16. Lower bound ( r = 2 ) We consider the greedy strategy: color all vertices red if feasible, blue otherwise.  after the losing move, G n, p contains a blue copy of F, every vertex of which would close a red copy of F. For F = K 4, e.g. or

17. Lower bound ( r = 2 )  Painter is safe if G n, p contains no such ‚dangerous‘ graphs. Lemma Among all dangerous graphs, F * is the one with minimal average degree, i.e., m ( F *) % m ( D ) for all dangerous graphs D. F*F* D

18. Lower bound ( r = 2 ) Corollary Let F be a clique or a cycle of arbitrary size. Playing greedily, Painter a.a.s. wins the online vertex- coloring game w.r.t. F and with two available colors if F *

19. Lower bound ( r = 3 ) Corollary Let F be a clique or a cycle of arbitrary size. Playing greedily, Painter a.a.s. wins the online vertex- coloring game w.r.t. F and with three available colors if F 3*F 3* F *

20. Lower bound Corollary Let F be a clique or a cycle of arbitrary size. Playing greedily, Painter a.a.s. wins the online vertex- coloring game w.r.t. F and with r R 2 available colors if …

21. The general case In general, it is smarter to greedily avoid a suitably chosen subgraph H of F instead of F itself.  general threshold function for game with r colors is where Maximization over r possibly different subgraphs H i  F, corresponding to a „smart greedy“ strategy. F H

22. A surprising example F = H 1 ] H 2 H1H1 H2H2 (lower bound only)

23. Upper bound Let p ( n ) [ p 0 ( F, r, n ) be given. We need to show: The probability that Painter can color all vertices of G n, p tends to 0 as n  , regardless of her strategy. Proof strategy: two-round exposure & induction on r First round n / 2 vertices, Painter may see them all at once use known offline results Second round remaining n / 2 vertices Due to coloring of first round, for many vertices one color is excluded  induction.

24. Upper bound V1V1 V2V2 F ° 1)Suppose Painter‘s offline-coloring of V 1 creates many (w.l.o.g.) red copies of F ° 2)Depending on the edges between V 1 and V 2, these copies induce a set Base ( R ) 4 V 2 of vertices that cannot be colored red. 3)Edges between vertices of Base ( R ) are independent of 1) and 2)  Base ( R ) induces a binomial random graph Base ( R ) F  need to show: Base ( R ) is large enough for induction hypothesis to be applicable.

25. There are a.a.s. many monochromatic copies of F ‘° in V 1 provided that work (Janson, Chernoff,...)  These induce enough vertices in (w.l.o.g.) Base ( R ) such that the induction hypothesis is applicable to the binomial random graph induced by Base ( R ). Upper bound

26. Main result (full) Theorem (Marciniszyn, S.; SODA ’07) Let F be a graph for which at least one F ° satisfies Then the threshold for the online vertex-coloring game w.r.t. F and with r R 1 colors is This threshold formula is not true for arbitrary graphs F ! F°F°

27. Intermission… (Questions?)

29. The online edge-coloring game Rules: one player, called Painter start with empty graph on n vertices edges appear u.a.r. one by one and have to be colored instantly (‚online‘) either red or blue game ends when monochromatic triangle appears Question: How many edges can Painter color? Theorem (Friedgut, Kohayakawa, Rödl, Ruci ń ski, Tetali, 2003): The threshold for this game is N 0 (n) = n 4/3. (easy, not main result of paper)

30. Back to online edge colorings Threshold is given by appearance of F *, yields threshold formula similarly to vertex case. Lower bound: Much harder to deal with overlapping outer copies! Works for arbitrary number of colors. Upper bound: Two-round exposure as in vertex case But: unclear how to setup an inductive argument to deal with r R 3 colors. F_F_ F °F ° F *

31. Online edge colorings Theorem (Marciniszyn, S., Steger, 2009) Let F be a graph that is not a tree, for which at least one F _ satisfies Then the threshold for the online edge-coloring game w.r.t. F and with two colors is F_F_