Online Ramsey theory for a triangle on FF-free graphs

Given a class $\mathcal{C}$ of graphs and a fixed graph $H$, the online Ramsey game for $H$ on $\mathcal C$ is a game between two players Builder and Painter as follows: an unbounded set of vertices is given as an initial state, and on each turn Builder introduces a new edge with the constraint that the resulting graph must be in $\mathcal C$, and Painter colors the new edge either red or blue. Builder wins the game if Painter is forced to make a monochromatic copy of $H$ at some point in the game. Otherwise, Painter can avoid creating a monochromatic copy of $H$ forever, and we say Painter wins the game. We initiate the study of characterizing the graphs $F$ such that for a given graph $H$, Painter wins the online Ramsey game for $H$ on $F$-free graphs. We characterize all graphs $F$ such that Painter wins the online Ramsey game for $C_3$ on the class of $F$-free graphs, except when $F$ is one particular graph. We also show that Painter wins the online Ramsey game for $C_3$ on the class of $K_4$-minor-free graphs, extending a result by Grytczuk, Ha{\l}uszczak, and Kierstead.Comment: 20 pages, 10 page

Badanie eksperymentalne podstawowych właściwości mechanicznych utwardzonego zaczynu gipsowego modyfikowanego dodatkiem mikroziaren polioksymetylenu

The development of the construction industry and the growing ecological awareness of society encourages us search for new solutions to improve building materials. Therefore, an attempt was made to improve building gypsum by modifying it with the addition of polyoxymethylene (POM). Polymer grains, with a particle size below and above 2 mm, were added to the samples in the amount of 1% and 2% relative to gypsum. The work contains the results of bending and compressive strength tests of prepared gypsum beams. It was shown that the compressive strength increased by 7% and the bending strength increased by 31% when compared to the reference test without the addition of polymer. All the obtained gypsum composites were characterized by a growth of strength. The best results were obtained for the sample containing gypsum composite modified with polymer in the amount of 1% and with a diameter of grains below 2 mm.Głównym celem pracy było sprawdzenie możliwości wykorzystania odpadów polioksymetylenowych (POM) w celu zwiększenia wytrzymałości na zginanie i ściskanie zaprawy gipsowej. Autorzy założyli, że polioksymetylen znacznie poprawi właściwości mechaniczne modyfikowanego gipsu. Ponadto przyjęto w badaniach, że wielkość ziaren polimeru może mieć wpływ na właściwości mechaniczne gipsu. W literaturze jest niewiele badań dotyczących modyfikacji materiałów budowlanych z dodatkiem polioksymetylenu. Większość z nich dotyczy jednak betonu. W związku z tym istnieje potrzeba zbadania wpływu POM na właściwości gipsu

On-line Ramsey numbers for paths and stars

We study on-line version of size-Ramsey numbers of graphs defined via a game played between Builder and Painter: in one round Builder joins two vertices by an edge and Painter paints it red or blue. The goal of Builder is to force Painter to create a monochromatic copy of a fixed graph H in as few rounds as possible. The minimum number of rounds (assuming both players play perfectly) is the on-line Ramsey number r(H) of the graph H. We determine exact values of r(H) for a few short paths and obtain a general upper bound r(Pn) ≤ 4n-7. We also study asymmetric version of this parameter when one of the target graphs is a star Sn with n edges. We prove that r(Sn,H)≤n ·e(H) when H is any tree, cycle or clique

Characterizing a social bookmarking and tagging network

Social networks and collaborative tagging systems are rapidly gaining popularity as a primary means for storing and sharing data among friends, family, colleagues, or perfect strangers as long as they have common interests. del.icio.us is a social network where people store and share their personal bookmarks. Most importantly, users tag their bookmarks for ease of information dissemination and later look up. However, it is the friendship links, that make delicious a social network. They exist independently of the set of bookmarks that belong to the users and have no relation to the tags typically assigned to the bookmarks. To study the interaction among users, the strength of the existing links and their hidden meaning, we introduce implicit links in the network. These links connect only highly “similar” users. Here, similarity can reflect different aspects of the user’s profile that makes her similar to any other user, such as number of shared bookmarks, or similarity of their tags clouds. We investigate the question whether friends have common interests, we gain additional insights on the strategies that users use to assign tags to their bookmarks, and we demonstrate that the graphs formed by implicit links have unique properties differing from binomial random graphs or random graphs with an expected power-law degree distribution

Clearing Connections by Few Agents

We study the problem of clearing connections by agents placed at some vertices in a directed graph. The agents can move only along directed paths. The objective is to minimize the number of agents guaranteeing that any pair of vertices can be connected by a underlying undirected path that can be cleared by the agents. We provide several results on the hardness, approximability and parameterized complexity of the problem. In particular, we show it to be: NP-hard, 2-approximable in polynomial-time, and solvable exactly in O(alpha n^3 2^{2alpha}) time, where alpha is the number of agents in the solution. In addition, we give a simple linear-time algorithm optimally solving the problem in digraphs whose underlying graphs are trees. Finally, we discuss a related problem, where the task is to clear with a minimum number of agents a subgraph of the underlying graph containing its spanning tree. We show that this problem also admits a 2-approximation in polynomial time