Games on Graphs and Other Combinatorial Problems

In this dissertation, we consider various combinatorial problems. The four chapters after the Introduction concern games on graphs, while latter on, we make progress on some questions in the settings of Rademacher sums and graph theory. In Chapter 2, we study the $(m,b)$ Maker-Breaker percolation game. This game, played by two players on the square lattice, was introduced by Day and Falgas-Ravry. The outcome of this game depends crucially on the parameters $m$ and $b$. Day and Falgas-Ravry showed that Breaker wins whenever $b \geq 2m$, but their approach then faces a barrier. We introduce a new, more global approach to study this game and to improve their results: we show that Breaker can in fact guarantee victory whenever $b \geq (2-1/14+o(1))m$. We also show that Breaker can win very fast in a different variant of this game as long as $b \geq 2m$. %We show that Breaker wins whenever $b \geq (2-1/14+o(1))m$, improving the results of Day and Falgas-Ravry and breaking certain barrier arising in their approach. We also show that Breaker can win very fast in a different variant of this game as long as $b \geq 2m$. In Chapters 3 and 4, we look at the Waiter-Client $K_k$-factor game, first studied by Clemens et al. Here, it is known that Waiter wins, and the question is how long the game will last if Waiter aims to win as fast as possible, Client tries to delay her as much as possible, and both players play optimally. In Chapter 3, we determine the duration of the game under the optimal play of both players when $k=3$, resolving the conjecture of Clemens et al. After that, we study the game for large $k$ in Chapter 4, and obtain the first known non-trivial lower bound for its duration in this case. In Chapter 5, we consider the so-called restricted online Ramsey numbers, which correspond to a certain colouring game in the Builder-Painter setup. We provide a tight lower bound for the restricted online Ramsey numbers of matchings as long as the number of the allowed colours is small, resolving the conjecture of Briggs and Cox. The setting in the next two chapters is the following. Set $X=\sum_{i=1}^{n} a_i \varepsilon_i$, where $\varepsilon_i$ are Rademacher random variables, i.e.~independent, identically distributed random variables taking values $\pm 1$ with probabilities $1/2$ each, and $\lbrace a_i \rbrace$ are arbitary real numbers. In Chapter 6, we make progress towards an old conjecture of Tomaszewski, which concerns concentration of such random variables $X$. In Chapter 7, we study the reverse problem and build up a framework that allows us to show anti-concentration results for Rademacher sums, and in turn we significantly improve the known results in this setting. In Chapter 8, we obtain the best possible bounds for the following problem, first studied by Erd\H{o}s, Pach, Pollak and Tuza: given a connected, triangle-free graph on $n$ vertices and of minimum degree at least $\delta$, how large can the radius of such a graph be? We also study the variant of this problem in which the triangle-free condition is replaced by a condition about the girth of our graph. In Chapter 9, we construct $P_n$-induced-saturated graph for each $n \geq 6$, answering the question of Axenovich and Csik\'os. Finally, in Chapter 10, we obtain a result about the existence of the antipodal paths with few colour changes in a two-colouring of the edges of the hypercube graph

Simulation-based assessment of data center waste heat utilization using aquifer thermal energy storage of a university campus

The global energy consumption of data centers (DCs) has experienced exponential growth over the last decade, that is expected to continue in the near future. Reasonable utilization of DC waste heat, which is dissipated during the computational process, can potentially be an effective solution to mitigate the environmental impact. However, the practical implementation of waste heat utilization in the DC environment is a very challenging task. The possible benefits of waste heat utilization are uncertain and difficult to quantify with the methods that are common in practice. This paper introduces a feasibility study in which dynamic simulation tools were used to predict the energy performance of a university campus resulting from the integration of a proposed DC system with an existing aquifer thermal energy storage (ATES). The presented study utilizes building energy simulation (BES) to evaluate uncertainty of the waste heat potential associated to various thermal management strategies of the proposed DC. Further in the feasibility study, the carbon footprint of the integrated approach is assessed for both the current and future situation based on measured data from the existing university campus and its district ATES system

Oral Microbiota Composition and Antimicrobial Antibody Response in Patients with Recurrent Aphthous Stomatitis

Recurrent aphthous stomatitis (RAS) is the most common disease of the oral mucosa, and it has been recently associated with bacterial and fungal dysbiosis. To study this link further, we investigated microbial shifts during RAS manifestation at an ulcer site, in its surroundings, and at an unaffected site, compared with healed mucosa in RAS patients and healthy controls. We sampled microbes from five distinct sites in the oral cavity. The one site with the most pronounced differences in microbial alpha and beta diversity between RAS patients and healthy controls was the lower labial mucosa. Detailed analysis of this particular oral site revealed strict association of the genus Selenomonas with healed mucosa of RAS patients, whereas the class Clostridia and genera Lachnoanaerobaculum, Cardiobacterium, Leptotrichia, and Fusobacterium were associated with the presence of an active ulcer. Furthermore, active ulcers were dominated by Malassezia, which were negatively correlated with Streptococcus and Haemophilus and positively correlated with Porphyromonas species. In addition, RAS patients showed increased serum levels of IgG against Mogibacterium timidum compared with healthy controls. Our study demonstrates that the composition of bacteria and fungi colonizing healthy oral mucosa is changed in active RAS ulcers, and that this alteration persists to some extent even after the ulcer is healed

An Overview of Cell-Based Assay Platforms for the Solute Carrier Family of Transporters

The solute carrier (SLC) superfamily represents the biggest family of transporters with important roles in health and disease. Despite being attractive and druggable targets, the majority of SLCs remains understudied. One major hurdle in research on SLCs is the lack of tools, such as cell-based assays to investigate their biological role and for drug discovery. Another challenge is the disperse and anecdotal information on assay strategies that are suitable for SLCs. This review provides a comprehensive overview of state-of-the-art cellular assay technologies for SLC research and discusses relevant SLC characteristics enabling the choice of an optimal assay technology. The Innovative Medicines Initiative consortium RESOLUTE intends to accelerate research on SLCs by providing the scientific community with high-quality reagents, assay technologies and data sets, and to ultimately unlock SLCs for drug discovery