Online Vertex Colorings of Random Graphs Without Monochromatic Subgraphs

1. Online Vertex Colorings of Random Graphs Without Monochromatic Subgraphs Reto Spöhel, ETH Zurich Joint work with Martin Marciniszyn

2. Introduction • Chromatic Number: Minimum number of colors needed to color vertices of a graph such that no two adjacent vertices have the same color. • Generalization: Instead of monochromatic edges, forbid monochromatic copies of some other fixed graph F. • Question: When are the vertices of a graph colorable with r colorswithout creating a monochromatic copy of some fixed graph F ? • For random graphs: solved in full generality by Luczak, Rucinski, Voigt, 1992 F=K3,r = 2

3. Introduction • ‚solved in full generality‘: Explicit threshold functionp0(F , r, n) such that • In fact, p0(F , r, n) = p0(F , n), i.e., the threshold does not depend on the number of colors r (!) • The threshold behaviour is even sharper than shown here. • We transfer this result into an online setting, where the vertices of Gn, phave to be colored one by one, without seeing the entire graph.

4. Introduction: our results • Explicit threshold functions p0(F , r, n)for online-colorability with rR2 colors for a large class of forbidden graphs F , including cliques and cycles of arbitrary size. • Unlike in the offline case, these thresholds • depend on the number of colors r • are coarse.

5. Introduction: related work • Question first considered for the analogous online edge-coloring (‚Ramsey‘) problem • Friedgut, Kohayakawa, Rödl, Rucinski, Tetali, 2003:F=K3,r = 2 • Marciniszyn, S., Steger, 2005+:F e.g. a clique or a cycle, r = 2 • Theory similar for edge- and vertex-colorings, but edge case is considerably more involved.

6. The online vertex-coloring game • Rules: • one player, called Painter • random graph Gn, p, initially hidden • vertices are revealedone by one along with induced edges • vertices have to be instantly (‚online‘) colored with one of rR2 available colors. • game ends as soon as Painter closes a monochromatic copy of some fixed forbidden graph F. • Question: • How dense can the underlying random graph be such that Painter can color allvertices a.a.s.?

7. Example F=K3,r = 2

8. Main result • Theorem (Marciniszyn, S., 2006+) 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 rR2 available colors is i.e.,

9. Bounds from ‚offline‘ graph properties • Gn, pcontains no copy of F • Painter wins with any strategy • Gn, p allows no r-vertex-coloring avoiding F •  Painter loses with any strategy •  the thresholds of these two ‚offline‘ graph properties bound p0(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 ‚Gn, pcontains 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 (Luczak, Rucinski, Voigt, 1992) Let F bea graph and let rR2. The threshold for the graph property ‚every r-vertex-coloring of Gn, 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) • . is also the threshold for the property ‚There are more than n copies of FinGn, p ‘ • Intuition: For p[p0 , the copies of Foverlap in vertices, and coloring Gn, p becomes difficult.

14. Main result revisited • 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 Fand with rR1available colors is • r=1 Small Subgraphs • r   exponent tends to exponent for offlinecase

15. Lower bound (r = 2) • Let p(n)/p0(F, 2, n) be given. We need to show: • There is a strategy which allows Painter to color all vertices of Gn, p a.a.s. • We consider the greedy strategy: color all vertices red if feasible, blue otherwise. • Proof strategy: • reduce the event that Painter fails to the appearance of a certain dangerous graph F * in Gn, p . • apply Small Subgraphs Theorem.

16. Lower bound (r = 2) • Analysis of the greedy strategy: • color all vertices red if feasible, blue otherwise. •  after the losing move, Gn, p contains a blue copy of F, every vertex of which would close a red copy of F. • For F = K4, e.g. or

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

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 * F3*

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 rR2 available colors if …

21. Upper bound • Let p(n)[p0(F, r, n) be given. We need to show: • The probability that Painter can color all vertices of Gn, p tends to 0 as n , regardless of her strategy. • Proof strategy: two-round exposure & induction onr • 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.

22. Upper bound F F ° V1 • Painter‘s offline-coloring of V1 creates many (w.l.o.g.) red copies of F ° • Depending on the edges between V1 and V2, these copies induce a set Base(R)4V2 of vertices thatcannot be colored red. • Edges between vertices of Base(R) are independent of 1) and 2) • Base(R) induces a binomial random graph Base(R) V2  need to show: Base(R) is large enough for induction hypothesis to be applicable.

23. Upper bound • There are a.a.s. many monochromatic copies ofF‘° in V1 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).

24. Generalization • In general, it is smarter to greedily avoid a suitably chosensubgraph H of F instead of F itself. •  general threshold function for game with r colors is where • Maximization over r possibly different subgraphs HiF, corresponding to a „smart greedy“ strategy. • Proved as a lower bound in full generality. • Proved as an upper bound assuming

25. Thank you! Questions?

26. F_ F° F* Similarly: 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 inductiveargument to deal with r³ 3 colors.

27. Online edge colorings • Theorem (Marciniszyn, S., Steger, 2005+) Let F be a 2-balanced 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 *

28. Online vertex colorings • Theorem (Marciniszyn, S., 2006+) Let F be a 1-balanced graph for which at least one F ° satisfies Then the threshold for the online vertex-coloring game w.r.t. F and with rR1colors is F ° F *