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

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 colors without creating a monochromatic copy of some fixed graph F ? For random graphs: solved in full generality by Luczak, Rucinski, Voigt, 1992 F = K 3, r = 2

Introduction ‚solved in full generality‘: Explicit threshold function 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 (!) The threshold behaviour is even sharper than shown here. We transfer this result into an online setting, where the vertices of G n, p have to be colored one by one, without seeing the entire graph.

Introduction: our results Explicit threshold functions p 0 (F, r, n) for online- colorability with r R 2 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.

Introduction: related work Question first considered for the analogous online edge-coloring (‚Ramsey‘) problem Friedgut, Kohayakawa, Rödl, Rucinski, Tetali, 2003: F = K 3, 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.

The online vertex-coloring game Rules: one player, called Painter 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 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 all vertices a.a.s.?

Example F = K 3, r = 2

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 r R 2 available colors is i.e.,

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.

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

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)

Vertex-colorings of random graphs Theorem (Luczak, Rucinski, 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

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 F in G n, p ‘ Intuition: For p [ p 0, the copies of F overlap in vertices, and coloring G n, p becomes difficult.

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

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. 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 G n, p. apply Small Subgraphs Theorem.

Lower bound (r = 2) Analysis of 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

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

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 *

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 *

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 …

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.

Upper bound V1V1 V2V2 F ° 1)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.

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

Generalization 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. Proved as a lower bound in full generality. Proved as an upper bound assuming

Thank you! Questions?

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 inductive argument to deal with r ³ 3 colors. F*F* F_F_ F°F°

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_F_

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 r R 1 colors is F°F° F *