On the path-avoidance vertex-coloring game Torsten Mütze, ETH Zürich Joint work with 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

A deterministic two-player game Fixed: a graph F, an integer r R 2 Two players: Painter vs. Builder The board: a graph, initially no vertices In each step Builder adds a new vertex v and a number of edges leading from v to previously added vertices Painter colors v with one of r colors Painter’s goal: Avoid a monochromatic copy of F Builder’s goal: Enforce a monochromatic copy of F Density restriction d : Builder must adhere to for all subgraphs H = half the avg. degree of H Motivation? Later!, a real number d > 0 H, d =1.8 Example: F =, r =2

A deterministic two-player game Density restriction d Builder can enforce F monochromatically in finitely many steps Painter can avoid monochromatic copies of F indefinitely Define the online vertex-Ramsey density as Theorem [M., Rast, Spöhel SODA ‘11] : For any F and r is computable is rational infimum attained as minimum

Theorem [M., Rast, Spöhel SODA ‘11] : For any F and r is computable is rational infimum attained as minimum A deterministic two-player game …nor for the two edge-coloring analogues [Rödl, Ruci ń ski ‘93], [Kurek, Ruci ń ski ‘05], [Belfrage, M., Spöhel ‘11+] None of those three statements is known for the offline quantity [Rödl, Ruci ń ski ‘93] zloty prize money for [Kurek, Ruci ń ski ‘94]

Motivation: A probabilistic one-player game [ Ł uczak, Ruci ń ski, Voigt ‘92] : vertex-Ramsey properties of random graphs [Marciniszyn, Spöhel ‘10] transferred this problem into the following online game setting: Fixed: a graph F, an integer r R 2 One player: Painter versus Builder random graph G n, p In each step one vertex of G n, p is revealed together with all backward edges Painter colors it with one of r colors Painter's goal: Avoid a monochromatic copy of F Example: F =, r =2

Motivation: A probabilistic one-player game p = edge probability Threshold For any : there is a Painter strategy that succeeds whp. For any : every Painter strategy fails whp. A bridge between the two games Can bound the threshold by designing and analyzing strategies for Painter and Builder in the deterministic game Theorem [M., Rast, Spöhel SODA ‘11] : For any fixed F and r, the threshold of the probabilistic one-player game is Painter vs. random graph Painter vs. Builder

Closed formulas for ? General algorithm to compute is rather complex; no hint how this quantity behaves for natural families of graphs F Theorem [Marciniszyn, Spöhel ‘10] : If F is (e.g.) a clique, a cycle, a complete bipartite graph, a hypercube, a wheel, or a star, then we have For those graphs, a simple greedy strategy is optimal for Painter: Colors 1,…, r ; greedily use highest-numbered color that does not complete a monochromatic copy of F (if impossible use color 1) Greedy witness graphs count edges and vertices of W Example: F = K 4, r =2 1 2 W

Closed formulas for ? This work:, a path on vertices Greedy strategy fails quite badly Online vertex-Ramsey density exhibits a surprisingly complex behavior vertices

Special case F = tree Lemma: Builder can enforce any tree F monochromatically without creating any cycles Focus on in the following T A convenient reparametrization: smallest k such that Builder can enforce F monochromatically without creating cycles and while building only trees T with

Our results Greedy lower bound: Theorem [M., Spöhel ’11] : Large values of : vertices count vertices of W W vertices Small values of :

Our results Approach via the asymmetric game Painter's goal: Avoid in color 1 and in color 2 Greedy lower bound: Theorem [M., Spöhel ’11] : For any fixed c there is a constant such that as c =1,2,3: c =4,5,6: We have irrational constants! count vertices of W W c vertices vertices greedy optimal

Proof ideas Painter strategy in the path-avoidance game = monotone walk in the integer lattice from (1,1) to x vertices y vertices use blue! x y the greedy strategy 1

Proof ideas Painter strategy in the path-avoidance game = monotone walk in the integer lattice from (1,1) to Evaluate the performance of the Painter strategy = recursive computation along the walk x y 1 s x := the size of the smallest component containing a blue P x sasa s x-1-a sbsb s y-1-b =1+s a +s x-1-a +s b +s y-1-b

Proof ideas Painter strategy in the path-avoidance game = monotone walk in the integer lattice from (1,1) to Evaluate the performance of the Painter strategy = recursive computation along the walk x y … Optimize over all strategies/walks (exponentially many!) Lemma: We have Algebraic problem of analyzing this recursion 1

Proof ideas What are the strategies/walks achieving the superquadratic lower bound ? Idea: Repeatedly nest scaled copies of an almost optimal walk for the -game self-similar fractal structure

Summary Path-avoidance vertex-coloring game: contrary to most other graph families (cliques, cycles, …), the greedy strategy fails quite badly the online vertex-Ramsey density exhibits a surprisingly complex behavior Thank you! Questions?