164 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    On Pairwise Graph Connectivity

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    A graph on at least k+1 vertices is said to have global connectivity k if any two of its vertices are connected by k independent paths. The local connectivity of two vertices is the number of independent paths between those specific vertices. This dissertation is concerned with pairwise connectivity notions, meaning that the focus is on local connectivity relations that are required for a number of or all pairs of vertices. We give a detailed overview about how uniformly k-connected and uniformly k-edge-connected graphs are related and provide a complete constructive characterization of uniformly 3-connected graphs, complementing classical characterizations by Tutte. Besides a tight bound on the number of vertices of degree three in uniformly 3-connected graphs, we give results on how the crossing number and treewidth behaves under the constructions at hand. The second central concern is to introduce and study cut sequences of graphs. Such a sequence is the multiset of edge weights of a corresponding Gomory-Hu tree. The main result in that context is a constructive scheme that allows to generate graphs with prescribed cut sequence if that sequence satisfies a shifted variant of the classical Erdős-Gallai inequalities. A complete characterization of realizable cut sequences remains open. The third central goal is to investigate the spectral properties of matrices whose entries represent a graph's local connectivities. We explore how the spectral parameters of these matrices are related to the structure of the corresponding graphs, prove bounds on eigenvalues and related energies, which are sums of absolute values of all eigenvalues, and determine the attaining graphs. Furthermore, we show how these results translate to ultrametric distance matrices and touch on a Laplace analogue for connectivity matrices and a related isoperimetric inequality

    Equitable coloring of planar graphs with maximum degree at least eight

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    The Chen-Lih-Wu Conjecture states that each connected graph with maximum degree Δ≥3\Delta\geq 3 that is not the complete graph KΔ+1K_{\Delta+1} or the complete bipartite graph KΔ,ΔK_{\Delta,\Delta} admits an equitable coloring with Δ\Delta colors. For planar graphs, the conjecture has been confirmed for Δ≥13\Delta\geq 13 by Yap and Zhang and for 9≤Δ≤129\leq \Delta\leq 12 by Nakprasit. In this paper, we present a proof that confirms the conjecture for graphs embeddable into a surface with non-negative Euler characteristic with maximum degree Δ≥9\Delta\geq 9 and for planar graphs with maximum degree Δ≥8\Delta\geq 8.Comment: 19 pages, 3 figure

    Integrality and cutting planes in semidefinite programming approaches for combinatorial optimization

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    Many real-life decision problems are discrete in nature. To solve such problems as mathematical optimization problems, integrality constraints are commonly incorporated in the model to reflect the choice of finitely many alternatives. At the same time, it is known that semidefinite programming is very suitable for obtaining strong relaxations of combinatorial optimization problems. In this dissertation, we study the interplay between semidefinite programming and integrality, where a special focus is put on the use of cutting-plane methods. Although the notions of integrality and cutting planes are well-studied in linear programming, integer semidefinite programs (ISDPs) are considered only recently. We show that manycombinatorial optimization problems can be modeled as ISDPs. Several theoretical concepts, such as the Chvátal-Gomory closure, total dual integrality and integer Lagrangian duality, are studied for the case of integer semidefinite programming. On the practical side, we introduce an improved branch-and-cut approach for ISDPs and a cutting-plane augmented Lagrangian method for solving semidefinite programs with a large number of cutting planes. Throughout the thesis, we apply our results to a wide range of combinatorial optimization problems, among which the quadratic cycle cover problem, the quadratic traveling salesman problem and the graph partition problem. Our approaches lead to novel, strong and efficient solution strategies for these problems, with the potential to be extended to other problem classes

    Approximate sampling and counting for spin models in graphs

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    En aquest treball abordem els problemes de mostreig i comptatge aproximat en models d'espins en grafs, recopilant els resultats més significatius de l'àrea i introduïnt els coneixements previs necessaris del camp de la física estadística. En particular, prestem especial atenció als mètodes generals de disseny d'algorismes desenvolupats per Weitz i Barvinok, així com els avenços recents en matèria de comptatge i mostreig de conjunts independents de mida donada. Així mateix, discutim com es podrien adaptar aquests arguments als problemes de comptatge i mostreig de coloracions amb les mides de cada color fixades, explicant amb detall la línia de recerca actual que estem duent a terme.En este trabajo abordamos los problemas de muestreo y conteo aproximado en modelos de espines en grafos, recopilando los resultados más significativos del campo e introduciendo el conocimiento previo necesario del área de la física estadística. En particular, prestamos especial atención a los métodos generales de diseño de algorismos desarrollados por Weitz y Barvinok, así como a los avances recientes en cuanto al conteo y muestreo de conjuntos independientes de un tamaño dado. Así mismo, discutimos cómo se podrían adaptar estos argumentos al problema de contar y muestrear coloraciones con el tamaño de cada color fijo, explicando en detalle la línea de investigación que estamos llevando a cabo actualmente.We approach the problems of approximate sampling and counting in spin models on graphs, surveying the most significant results in the area and introducing the necessary background from statistical physics. We pay particular attention to the general algorithm design frameworks developed by Weitz and Barvinok, as well as to the newer results on counting and sampling independent sets of given size. In addition, we discuss the adaptation of the arguments behind these results to count and sample colorings with fixed color sizes, explaining in detail the current research line we are undertaking.Outgoin

    Sliding into the Future: Investigating Sliding Windows in Temporal Graphs

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    Graphs are fundamental tools for modelling relations among objects in various scientific fields. However, traditional static graphs have limitations when it comes to capturing the dynamic nature of real-world systems. To overcome this limitation, temporal graphs have been introduced as a framework to model graphs that change over time. In temporal graphs the edges among vertices appear and disappear at specific time steps, reflecting the temporal dynamics of the observed system, which allows us to analyse time dependent patterns and processes. In this paper we focus on the research related to sliding time windows in temporal graphs. Sliding time windows offer a way to analyse specific time intervals within the lifespan of a temporal graph. By sliding the window along the timeline, we can examine the graph’s characteristics and properties within different time periods. This paper provides an overview of the research on sliding time windows in temporal graphs. Although progress has been made in this field, there are still many interesting questions and challenges to be explored. We discuss some of the open problems and highlight their potential for future research

    Graph Coverings with Few Eigenvalues or No Short Cycles

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    This thesis addresses the extent of the covering graph construction. How much must a cover X resemble the graph Y that it covers? How much can X deviate from Y? The main statistics of X and Y which we will measure are their regularity, the spectra of their adjacency matrices, and the length of their shortest cycles. These statistics are highly interdependent and the main contribution of this thesis is to advance our understanding of this interdependence. We will see theorems that characterize the regularity of certain covering graphs in terms of the number of distinct eigenvalues of their adjacency matrices. We will see old examples of covers whose lack of short cycles is equivalent to the concentration of their spectra on few points, and new examples that indicate certain limits to this equivalence in a more general setting. We will see connections to many combinatorial objects such as regular maps, symmetric and divisible designs, equiangular lines, distance-regular graphs, perfect codes, and more. Our main tools will come from algebraic graph theory and representation theory. Additional motivation will come from topological graph theory, finite geometry, and algebraic topology

    An analysis between different algorithms for the graph vertex coloring problem

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    This research focuses on an analysis of different algorithms for the graph vertex coloring problem. Some approaches to solving the problem are discussed. Moreover, some studies for the problem and several methods for its solution are analyzed as well. An exact algorithm (using the backtracking method) is presented. The complexity analysis of the algorithm is discussed. Determining the average execution time of the exact algorithm is consistent with the multitasking mode of the operating system. This algorithm generates optimal solutions for all studied graphs. In addition, two heuristic algorithms for solving the graph vertex coloring problem are used as well. The results show that the exact algorithm can be used to solve the graph vertex coloring problem for small graphs with 30-35 vertices. For half of the graphs, all three algorithms have found the optimal solutions. The suboptimal solutions generated by the approximate algorithms are identical in terms of the number of colors needed to color the corresponding graphs. The results show that the linear increase in the number of vertices and edges of the analyzed graphs causes a linear increase in the number of colors needed to color these graphs

    Ramsey multiplicity and the Tur\'an coloring

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    Extending an earlier conjecture of Erd\H{o}s, Burr and Rosta conjectured that among all two-colorings of the edges of a complete graph, the uniformly random coloring asymptotically minimizes the number of monochromatic copies of any fixed graph HH. This conjecture was disproved independently by Sidorenko and Thomason. The first author later found quantitatively stronger counterexamples, using the Tur\'an coloring, in which one of the two colors spans a balanced complete multipartite graph. We prove that the Tur\'an coloring is extremal for an infinite family of graphs, and that it is the unique extremal coloring. This yields the first determination of the Ramsey multiplicity constant of a graph for which the Burr--Rosta conjecture fails. We also prove an analogous three-color result. In this case, our result is conditional on a certain natural conjecture on the behavior of two-color Ramsey numbers.Comment: 37 page

    On the Total Set Chromatic Number of Graphs

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    Given a vertex coloring c of a graph, the neighborhood color set of a vertex is defined to be the set of all of its neighbors’ colors. The coloring c is called a set coloring if any two adjacent vertices have different neighborhood color sets. The set chromatic number χs(G) of a graph G is the minimum number of colors required in a set coloring of G. In this work, we investigate a total analog of set colorings; that is, we study set colorings of the total graph of graphs. Given a graph G = (V, E), its total graph T(G) is the graph whose vertex set is V ∪ E and in which two vertices are adjacent if and only if their corresponding elements in G are adjacent or incident. First, we establish sharp bounds for the set chromatic number of the total graph of a graph. Furthermore, we study the set colorings of the total graph of different families of graphs
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