946 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    On the use of senders for minimal Ramsey theory

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    This thesis investigates problems related to extremal and probabilistic graph theory. Our focus lies on the highly dynamic field of Ramsey theory. The foundational result of this field was proved in 1930 by Franck P. Ramsey. It implies that for every integer t and every sufficiently large complete graph Kn, every colouring of the edges of Kn with colours red and blue contains a red copy or a blue copy of Kt. Let q ⩾ 2 represent a number of colours, and let H1,..., Hq be graphs. A graph G is said to be q-Ramsey for the tuple (H1,...,Hq) if, for every colouring of the edges of G with colours {1, . . . , q}, there exists a colour i and a monochromatic copy of Hi in colour i. As we often want to understand the structural properties of the collection of graphs that are q-Ramsey for (H1,..., Hq), we restrict our attention to the graphs that are minimal for this property, with respect to subgraph inclusion. Such graphs are said to be q-Ramsey-minimal for (H1,..., Hq). In 1976, Burr, Erdős, and Lovász determined, for every s, t ⩾ 3, the smallest minimum degree of a graph G that is 2-Ramsey-minimal for (Ks, Kt). Significant efforts have been dedicated to generalising this result to a higher number of colours, q⩾3, starting with the ‘symmetric’ q-tuple (Kt,..., Kt). In this thesis, we improve on the best known bounds for this parameter, providing state-of-the-art bounds in different (q, t)-regimes. These improvements rely on constructions based on finite geometry, which are then used to prove the existence of extremal graphs with certain key properties. Another crucial ingredient in these proofs is the existence of gadget graphs, called signal senders, that were initially developed by Burr, Erdős, and Lovász in 1976 for pairs of complete graphs. Until now, these senders have been shown to exist only in the two-colour setting, when q = 2, or in the symmetric multicolour setting, when H1,..., Hq are pairwise isomorphic. In this thesis, we then construct similar gadgets for all tuples of complete graphs, providing the first known constructions of these tools in the multicolour asymmetric setting. We use these new senders to prove far-reaching generalisations of several classical results in the area

    Spectral pseudorandomness and the road to improved clique number bounds for Paley graphs

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    We study subgraphs of Paley graphs of prime order pp induced on the sets of vertices extending a given independent set of size aa to a larger independent set. Using a sufficient condition proved in the author's recent companion work, we show that a family of character sum estimates would imply that, as pp \to \infty, the empirical spectral distributions of the adjacency matrices of any sequence of such subgraphs have the same weak limit (after rescaling) as those of subgraphs induced on a random set including each vertex independently with probability 2a2^{-a}, namely, a Kesten-McKay law with parameter 2a2^a. We prove the necessary estimates for a=1a = 1, obtaining in the process an alternate proof of a character sum equidistribution result of Xi (2022), and provide numerical evidence for this weak convergence for a2a \geq 2. We also conjecture that the minimum eigenvalue of any such sequence converges (after rescaling) to the left edge of the corresponding Kesten-McKay law, and provide numerical evidence for this convergence. Finally, we show that, once a3a \geq 3, this (conjectural) convergence of the minimum eigenvalue would imply bounds on the clique number of the Paley graph improving on the current state of the art due to Hanson and Petridis (2021), and that this convergence for all a1a \geq 1 would imply that the clique number is o(p)o(\sqrt{p}).Comment: 43 pages, 1 table, 6 figure

    Terrain generation algorithms

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    Procedural terrain generation has become common in games as a whole and in indie games in particular. With procedural terrain generation developers can relatively easily create static or dynamically expanding game areas. Also it is more cost effective since large part of manual work can be automated which traditional game areas would require. Goal of this thesis is to introduce and evaluate different algorithms that are used or have potential use cases in terrain generation. Such algorithms as various noise functions, which are widely used in the realm of terrain generation, a number of dungeon algorithms, which use variety of methods to generate the dungeon, fractal algorithm, and volumetric terrain generation algorithm which uses a combination of noise and fractal algorithms. Algorithms and techniques will be searched from various scientific articles and literary sources. Metrics used for terrain generation algorithm evaluation will also be introduced, and algorithms in this thesis will be evaluated using these metrics. During evaluation it was noticed that the evaluated noise functions are generally capable of runtime terrain generation, but are lacking in customization and control since parameters are usually related to the algorithm rather than the resulting terrain. Albeit these shortcomings both Perlin and Simplex noise stand out for their ability to generate good quality terrains. On the other hand most of the evaluated dungeon generation algorithms are incapable of generating terrain during runtime with few exceptions. Also guaranteeing connectivity of rooms or areas in dungeon can be challenge in some algorithms. The introduced fractal algorithm is metrics wise similar to Perlin and Simplex noise even though it uses completely different method to generate the terrain. The volumetric terrain generation algorithm is the only algorithm capable of generating volumetric terrain and its high level of parametrization and customization is its strongest quality

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Pairwise disjoint perfect matchings in rr-edge-connected rr-regular graphs

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    Thomassen [Problem 1 in Factorizing regular graphs, J. Combin. Theory Ser. B, 141 (2020), 343-351] asked whether every rr-edge-connected rr-regular graph of even order has r2r-2 pairwise disjoint perfect matchings. We show that this is not the case if r2 mod 4r \equiv 2 \text{ mod } 4. Together with a recent result of Mattiolo and Steffen [Highly edge-connected regular graphs without large factorizable subgraphs, J. Graph Theory, 99 (2022), 107-116] this solves Thomassen's problem for all even rr. It turns out that our methods are limited to the even case of Thomassen's problem. We then prove some equivalences of statements on pairwise disjoint perfect matchings in highly edge-connected regular graphs, where the perfect matchings contain or avoid fixed sets of edges. Based on these results we relate statements on pairwise disjoint perfect matchings of 5-edge-connected 5-regular graphs to well-known conjectures for cubic graphs, such as the Fan-Raspaud Conjecture, the Berge-Fulkerson Conjecture and the 55-Cycle Double Cover Conjecture.Comment: 24 page

    Cluster expansion methods in rigorous statistical mechanics

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    This draft is intended to be used as class notes for a grad course on rigorous statistical mechanics at math department of UFMG. It should be considered as a very prelimivary version and a work in progress. Several chapters lack references, exercises, and revision

    Time-resolved spectroscopy of two-dimensional systems: from the conventional method to a novel cavity-enhanced solution

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    The work presented in this dissertation is dedicated to the characterization of the excited state dynamics of thin films through time-resolved spectroscopy, with emphasis on developing a methodology that is able to resolve weak transient absorption signals from optically thin films. With this aim, the conventional transient absorption spectroscopy method is first utilized to characterize semiconducting monolayers and organic nanosheet semiconductors. Although these are physically thin, they present relatively strong transient absorption signals of a few mOD (units of optical density), which allows to characterize their excited state dynamics with the conventional machinery, not needing further signal enhancements or complex noise-minimizing techniques. Nonetheless, the former does not represent the reality of detecting the transient photoexcited dynamics of few-layered systems. For this reason, the last chapter of this thesis introduces a new approach for the sensitive detection of two-dimensional samples with marginal molecular extinction coefficients: A novel methodology that multiplies the interaction length of the light with the sample, designated cavity ring-down transient absorption spectroscopy (CRD-TAS). Being at the present time in the midst of its development, the prospect efficiency and working capabilities of the novel CRD-TAS technique are hereby evaluated, and the strategies for further improvements are discussed

    On uniquely packable trees

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    An ii-packing in a graph GG is a set of vertices that are pairwise distance more than ii apart. A \emph{packing colouring} of GG is a partition X={X1,X2,,Xk}X=\{X_{1},X_{2},\ldots,X_{k}\} of V(G)V(G) such that each colour class XiX_{i} is an ii-packing. The minimum order kk of a packing colouring is called the packing chromatic number of GG, denoted by χρ(G)\chi_{\rho}(G). In this paper we investigate the existence of trees TT for which there is only one packing colouring using χρ(T)\chi_\rho(T) colours. For the case χρ(T)=3\chi_\rho(T)=3, we completely characterise all such trees. As a by-product we obtain sets of uniquely 33-χρ\chi_\rho-packable trees with monotone χρ\chi_{\rho}-coloring and non-monotone χρ\chi_{\rho}-coloring respectively

    A survey of parameterized algorithms and the complexity of edge modification

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    The survey is a comprehensive overview of the developing area of parameterized algorithms for graph modification problems. It describes state of the art in kernelization, subexponential algorithms, and parameterized complexity of graph modification. The main focus is on edge modification problems, where the task is to change some adjacencies in a graph to satisfy some required properties. To facilitate further research, we list many open problems in the area.publishedVersio
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