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- Josef Cibulka, Pu Gao, Marek Krcál, Tomás Valla, Pavel Valtr
- Discrete & Computational Geometry
- 2015

We prove polynomial upper bounds of geometric Ramsey numbers of pathwidth2 outerplanar triangulations in both convex and general cases. We also prove that the geometric Ramsey numbers of the ladder graph on 2n vertices are bounded by O(n3) and O(n10), in the convex and general case, respectively. We then apply similar methods to prove an nO(log(n)) upper… (More)

- Jaroslav Nesetril, Tomás Valla
- Journal of Graph Theory
- 2010

Beck introduced the concept of Ramsey games by studying the game versions of Ramsey and van der Waerden theorems. We contribute to this topic by investigating games corresponding to structural extensions of Ramsey and van der Waerden theorems—the theorem of Brauer, structural and restricted Ramsey theorems.

- Georgios Piliouras, Tomás Valla, László A. Végh
- Theory of Computing Systems
- 2012

We design a new class of vertex and set cover games, where the price of anarchy bounds match the best known constant factor approximation guarantees for the centralized optimization problems for linear and also for submodular costs. This is in contrast to all previously studied covering games, where the price of anarchy grows linearly with the size of the… (More)

- Robert Sámal, Rudolf Stolar, Tomás Valla
- IWOCA
- 2011

The guarding game is a game in which a set of cops has to guard a region in a digraph against a robber. The robber and the cops are placed on vertices of the graph; they take turns in moving to adjacent vertices (or staying). The goal of the robber is to enter the guarded region at a vertex with no cop on it. The problem is to find the minimum number of… (More)

Searching in partially ordered structures has been considered in the context of information retrieval and efficient tree-like indexes, as well as in hierarchy based knowledge representation. In this paper we focus on tree-like partial orders and consider the problem of identifying an initially unknown vertex in a tree by asking edge queries: an edge query e… (More)

- Vít Jelínek, Jan Kyncl, Rudolf Stolar, Tomás Valla
- Combinatorica
- 2009

We prove that for any partition of the plane into a closed set C and an open set O and for any configuration T of three points, there is a translated and rotated copy of T contained in C or in O. Apart from that, we consider partitions of the plane into two sets whose common boundary is a union of piecewise linear curves. We show that for any such partition… (More)

- Pavel Dvořák, Tomáš Valla
- 2016

Consider a hypergraph H n where the vertices are points of the d-dimensional combinatorial cube n and the edges are all sets of n points such that they are in one line. We study the structure of the group of automorphisms of H n, i.e., permutations of points of n d preserving the edges. In this paper we provide a complete characterization. Moreover, we… (More)

- Robert Sámal, Tomás Valla
- ArXiv
- 2011

The guarding game is a game in which several cops try to guard a region in a (directed or undirected) graph against a robber. The robber and the cops are placed on the vertices of the graph; they take turns in moving to adjacent vertices (or staying), cops inside the guarded region, the robber on the remaining vertices (the robber-region). The goal of the… (More)

- Pavel Dvorák, Tomás Valla
- Electronic Notes in Discrete Mathematics
- 2015

An online Ramsey game (G, H) is a game between Builder and Painter, alternating in turns. In each round Builder draws an edge and Painter colors it either red or blue. Builder wins if after some round there is a monochromatic copy of the graph H, otherwise Painter is the winner. The rule for Builder is that after each his move the resulting graph must… (More)

- Z. Dvořák, R. Škrekovski, T. Valla
- 2006

Let G be a graph and let c : V (G) → ({1,...,5} 2 ) be an assignment of 2-element subsets of the set {1, . . . , 5} to the vertices of G such that for every edge vw, the sets c(v) and c(w) are disjoint. We call such an assignment a (5, 2)-coloring. A graph is (5,2)-colorable if and only if it has a homomorphism to the Petersen graph. The odd-girth of a… (More)