Probabilistic one-player vertex-coloring games via deterministic two-player games The deterministic game Torsten Mütze, ETH Zürich Joint work with Thomas Rast and Reto Spöhel (MPI Saarbrücken) TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A AA A A

Ramsey properties of random graphs Problem: Color the vertices of a graph G with r colors without creating a monochromatic copy of F F =, r = 2 G G has a valid coloring K5K5 Explicit threshold function Solved in full generality for random graph G n, p [Luczak, Rucinski, Voigt ’92]: For we have where

The online setting One player-game: Painter vs. random graph Reveal vertices of hidden G n, p one by one with induced edges Painter assigns one of r colors to each vertex immediately Goal: avoid a monochromatic copy of F F =, r = 2 Threshold there is a strategy that succeeds with probability 1-o(1) every strategy fails with probability 1-o(1) p ( n ) = edge probability of G n, p

Painter vs. random graph – previous results Greedy strategy: use highest available color from {1,..., r } that does not complete a monochromatic copy of F [Marciniszyn, Spöhel SODA ’07]: Greedy strategy is optimal for large class of graphs F, including cliques and cycles (=> explicit threshold functions) Greedy strategy optimal ? This work: combinatorial characterization of the threshold => for any F and r, compute s.t.

Painter vs. random graph Builder d Builder can enforce F monochromatically in finitely many steps Painter can avoid monochromatic copies of F indefinitely Definition: Online vertex-Ramsey density Adversary Builder adds vertices and backward edges For some fixed real number d (density restriction): board

Painter vs. Builder Painter vs. random graph Theorem 1 [M., Rast, Spöhel SODA ’11]: For any F and r is computable is rational infimum attained as minimum Theorem 2 [M., Rast, Spöhel SODA ’11]: For any F and r, the threshold of the probabilistic one-player game is Reduced probabilistic problem to purely combinatorial problem Thomas’ talk on Thursday This talk

Painter vs. Builder – Remarks Theorem 1 [M., Rast, Spöhel SODA ’11]: For any F and r is computable is rational infimum attained as minimum zloty prize money for Nor for the two edge coloring analogues [Kurek, Rucinski ‘05], [Belfrage, M., Spöhel ‘10+] None of those three statements known for the offline quantity [Kurek, Rucinski ’94]

Computing – General remarks Straightforward implementation computes for any F with in under 10 minutes (recall that R ( K 5 ) is unknown!) function involves optimization over all reasonable Painter strategies How we compute : = complicated function of the return values of a complicated algorithm A simpler general formula? Greedy strategy optimal e.g. If we knew: Builder never needs more than steps => Theorem 1 ( computable, rational, infimum a minimum)

A simpler general formula? Observation: Builder can enforce any tree F monochromatically and not close any cycles B  tree size restriction k density restriction Special case: F =tree smallest k s.t. Builder can enforce F with tree size restriction k 

A simpler general formula? vertices greedy lower bound Theorem 3 [M., Spöhel ’10+] : Asymptotics for large ? Greedy strategy fails quite badly smallest k s.t. Builder can enforce F with tree size restriction k  Probably no!

Basic observations Pigeonholing W.l.o.g. Builder always has as many copies as he needs -fraction Focus on monochromatic substructures  W.l.o.g. Builder always plays like this: ? “history” graph

Our algorithm to compute Idea: systematic exploration of Builder’s options for a fixed Painter strategy Determine smallest density restriction d for which Builder can enforce F In the end: maximization over all reasonable Painter strategies

In each color: maintain list of enforced subgraphs of F Keep computing new entries for the lists Our algorithm to compute Problem: No obvious termination criterion! Idea: systematic exploration of Builder’s options for a fixed Painter strategy {,, … } {,,, …} Color 1 Color 2

Magic: history graphs vertex-weighted graphs (weight=encoded density information) Our algorithm to compute {,,, …} Color Weights depend only on the vertex ordering (very briefly: minimize rather than maximizing ) We compute vertex weights by dynamic programming over vertex-ordered subgraphs of F  smallest density restriction to enforce F for a fixed Painter strategy

Thank you! Questions? more in Thomas’ talk on Thursday