19 research outputs found

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Clase de pesos multilineales asociados a propiedades de continuidad de conmutadores de operadores fraccionarios generalizados

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    Estudiamos propiedades de continuidad para conmutadores de orden superior asociados a operadores fraccionarios generalizados que resultan ser una extensión del operador integral fraccionario IalphamI_alpha^m en el contexto multilineal. Las acotaciones son entre un producto de espacios de Lebesgue pesados y ciertos espacios de tipo Lipschitz pesados, extendiendo estimaciones previas de la literatura para el caso lineal. Este estudio incluye dos tipos diferentes de conmutadores y condiciones suficientes en los pesos involucrados para garantizar las acotaciones referidas anteriormente. También se incluye el rango óptimo de los parámetros involucrados, que se entiende en el sentido de describir una región fuera de la cual los pesos son triviales. El análisis incluye también ejemplos de pesos que abarcan esta región de optimalidad.Fil: Recchi, Diana Jorgelina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; ArgentinaFil: Berra, Fabio Martín. Universidad Nacional del Litoral. Facultad de Ingeniería Química. Departamento de Matemáticas; ArgentinaFil: Pradolini, Gladis Guadalupe. Universidad Nacional del Litoral. Facultad de Ingeniería Química. Departamento de Matemáticas; ArgentinaXVII Congreso Antonio MonteiroBahía BlancaArgentinaUniversidad Nacional del SurInstituto de Matemática de Bahía Blanc

    Streaming deletion problems parameterized by vertex cover

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    Streaming is a model where an input graph is provided one edge at a time, instead of being able to inspect it at will. In this work, we take a parameterized approach by assuming a vertex cover of the graph is given, building on work of Bishnu et al. [COCOON 2020]. We show the further potency of combining this parameter with the Adjacency List streaming model to obtain results for vertex deletion problems. This includes kernels, parameterized algorithms, and lower bounds for the problems of Π-free Deletion, H-free Deletion, and the more specific forms of Cluster Vertex Deletion and Odd Cycle Transversal. We focus on the complexity in terms of the number of passes over the input stream, and the memory used. This leads to a pass/memory trade-off, where a different algorithm might be favourable depending on the context and instance. We also discuss implications for parameterized complexity in the non-streaming setting

    LIPIcs, Volume 258, SoCG 2023, Complete Volume

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    LIPIcs, Volume 258, SoCG 2023, Complete Volum

    Pliability and approximating max-CSPs

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    We identify a sufficient condition, treewidth-pliability, that gives a polynomial-time algorithm for an arbitrarily good approximation of the optimal value in a large class of Max-2-CSPs parameterised by the class of allowed constraint graphs (with arbitrary constraints on an unbounded alphabet). Our result applies more generally to the maximum homomorphism problem between two rational-valued structures. The condition unifies the two main approaches for designing a polynomial-time approximation scheme. One is Baker’s layering technique, which applies to sparse graphs such as planar or excluded-minor graphs. The other is based on Szemer´edi’s regularity lemma and applies to dense graphs. We extend the applicability of both techniques to new classes of Max-CSPs. On the other hand, we prove that the condition cannot be used to find solutions (as opposed to approximating the optimal value) in general. Treewidth-pliability turns out to be a robust notion that can be defined in several equivalent ways, including characterisations via size, treedepth, or the Hadwiger number. We show connections to the notions of fractional-treewidth-fragility from structural graph theory, hyperfiniteness from the area of property testing, and regularity partitions from the theory of dense graph limits. These may be of independent interest. In particular we show that a monotone class of graphs is hyperfinite if and only if it is fractionallytreewidth-fragile and has bounded degree

    EUROCOMB 21 Book of extended abstracts

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    Rooted structures in graphs: a project on Hadwiger's conjecture, rooted minors, and Tutte cycles

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    Hadwigers Vermutung ist eine der anspruchsvollsten Vermutungen für Graphentheoretiker und bietet eine weitreichende Verallgemeinerung des Vierfarbensatzes. Ausgehend von dieser offenen Frage der strukturellen Graphentheorie werden gewurzelte Strukturen in Graphen diskutiert. Eine Transversale einer Partition ist definiert als eine Menge, welche genau ein Element aus jeder Menge der Partition enthält und sonst nichts. Für einen Graphen G und eine Teilmenge T seiner Knotenmenge ist ein gewurzelter Minor von G ein Minor, der T als Transversale seiner Taschen enthält. Sei T eine Transversale einer Färbung eines Graphen, sodass es ein System von kanten-disjunkten Wegen zwischen allen Knoten aus T gibt; dann stellt sich die Frage, ob es möglich ist, die Existenz eines vollständigen, in T gewurzelten Minors zu gewährleisten. Diese Frage ist eng mit Hadwigers Vermutung verwoben: Eine positive Antwort würde Hadwigers Vermutung für eindeutig färbbare Graphen bestätigen. In dieser Arbeit wird ebendiese Fragestellung untersucht sowie weitere Konzepte vorgestellt, welche bekannte Ideen der strukturellen Graphentheorie um eine Verwurzelung erweitern. Beispielsweise wird diskutiert, inwiefern hoch zusammenhängende Teilmengen der Knotenmenge einen hoch zusammenhängenden, gewurzelten Minor erzwingen. Zudem werden verschiedene Ideen von Hamiltonizität in planaren und nicht-planaren Graphen behandelt.Hadwiger's Conjecture is one of the most tantalising conjectures for graph theorists and offers a far-reaching generalisation of the Four-Colour-Theorem. Based on this major issue in structural graph theory, this thesis explores rooted structures in graphs. A transversal of a partition is a set which contains exactly one element from each member of the partition and nothing else. Given a graph G and a subset T of its vertex set, a rooted minor of G is a minor such that T is a transversal of its branch set. Assume that a graph has a transversal T of one of its colourings such that there is a system of edge-disjoint paths between all vertices from T; it comes natural to ask whether such graphs contain a minor rooted at T. This question of containment is strongly related to Hadwiger's Conjecture; indeed, a positive answer would prove Hadwiger's Conjecture for uniquely colourable graphs. This thesis studies the aforementioned question and besides, presents several other concepts of attaching rooted relatedness to ideas in structural graph theory. For instance, whether a highly connected subset of the vertex set forces a highly connected rooted minor. Moreover, several ideas of Hamiltonicity in planar and non-planar graphs are discussed

    Cliques, Degrees, and Coloring: Expanding the ω, Δ, χ paradigm

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    Many of the most celebrated and influential results in graph coloring, such as Brooks' Theorem and Vizing's Theorem, relate a graph's chromatic number to its clique number or maximum degree. Currently, several of the most important and enticing open problems in coloring, such as Reed's ω,Δ,χ\omega, \Delta, \chi Conjecture, follow this theme. This thesis both broadens and deepens this classical paradigm. In Part~1, we tackle list-coloring problems in which the number of colors available to each vertex vv depends on its degree, denoted d(v)d(v), and the size of the largest clique containing it, denoted ω(v)\omega(v). We make extensive use of the probabilistic method in this part. We conjecture the ``list-local version'' of Reed's Conjecture, that is every graph is LL-colorable if LL is a list-assignment such that L(v)(1ε)(d(v)+1)+εω(v))|L(v)| \geq \lceil (1 - \varepsilon)(d(v) + 1) + \varepsilon\omega(v))\rceil for each vertex vv and ε1/2\varepsilon \leq 1/2, and we prove this for ε1/330\varepsilon \leq 1/330 under some mild additional assumptions. We also conjecture the ``mad\mathrm{mad} version'' of Reed's Conjecture, even for list-coloring. That is, for ε1/2\varepsilon \leq 1/2, every graph GG satisfies \chi_\ell(G) \leq \lceil (1 - \varepsilon)(\mad(G) + 1) + \varepsilon\omega(G)\rceil, where mad(G)\mathrm{mad}(G) is the maximum average degree of GG. We prove this conjecture for small values of ε\varepsilon, assuming ω(G)mad(G)log10mad(G)\omega(G) \leq \mathrm{mad}(G) - \log^{10}\mathrm{mad}(G). We actually prove a stronger result that improves bounds on the density of critical graphs without large cliques, a long-standing problem, answering a question of Kostochka and Yancey. In the proof, we use a novel application of the discharging method to find a set of vertices for which any precoloring can be extended to the remainder of the graph using the probabilistic method. Our result also makes progress towards Hadwiger's Conjecture: we improve the best known bound on the chromatic number of KtK_t-minor free graphs by a constant factor. We provide a unified treatment of coloring graphs with small clique number. We prove that for Δ\Delta sufficiently large, if GG is a graph of maximum degree at most Δ\Delta with list-assignment LL such that for each vertex vV(G)v\in V(G), L(v)72d(v)min{ln(ω(v))ln(d(v)),ω(v)ln(ln(d(v)))ln(d(v)),log2(χ(G[N(v)])+1)ln(d(v))}|L(v)| \geq 72\cdot d(v)\min\left\{\sqrt{\frac{\ln(\omega(v))}{\ln(d(v))}}, \frac{\omega(v)\ln(\ln(d(v)))}{\ln(d(v))}, \frac{\log_2(\chi(G[N(v)]) + 1)}{\ln(d(v))}\right\} and d(v)ln2Δd(v) \geq \ln^2\Delta, then GG is LL-colorable. This result simultaneously implies three famous results of Johansson from the 90s, as well as the following new bound on the chromatic number of any graph GG with ω(G)ω\omega(G)\leq \omega and Δ(G)Δ\Delta(G)\leq \Delta for Δ\Delta sufficiently large: χ(G)72ΔlnωlnΔ.\chi(G) \leq 72\Delta\sqrt{\frac{\ln\omega}{\ln\Delta}}. In Part~2, we introduce and develop the theory of fractional coloring with local demands. A fractional coloring of a graph is an assignment of measurable subsets of the [0,1][0, 1]-interval to each vertex such that adjacent vertices receive disjoint sets, and we think of vertices ``demanding'' to receive a set of color of comparatively large measure. We prove and conjecture ``local demands versions'' of various well-known coloring results in the ω,Δ,χ\omega, \Delta, \chi paradigm, including Vizing's Theorem and Molloy's recent breakthrough bound on the chromatic number of triangle-free graphs. The highlight of this part is the ``local demands version'' of Brooks' Theorem. Namely, we prove that if GG is a graph and f:V(G)[0,1]f : V(G) \rightarrow [0, 1] such that every clique KK in GG satisfies vKf(v)1\sum_{v\in K}f(v) \leq 1 and every vertex vV(G)v\in V(G) demands f(v)1/(d(v)+1/2)f(v) \leq 1/(d(v) + 1/2), then GG has a fractional coloring ϕ\phi in which the measure of ϕ(v)\phi(v) for each vertex vV(G)v\in V(G) is at least f(v)f(v). This result generalizes the Caro-Wei Theorem and improves its bound on the independence number, and it is tight for the 5-cycle

    Hadwiger Numbers and Gallai-Ramsey Numbers of Special Graphs

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    This dissertation explores two separate topics on graphs. We first study a far-reaching generalization of the Four Color Theorem. Given a graph G, we use chi(G) to denote the chromatic number; alpha(G) the independence number; and h(G) the Hadwiger number, which is the largest integer t such that the complete graph K_t can be obtained from a subgraph of G by contracting edges. Hadwiger\u27s conjecture from 1943 states that for every graph G, h(G) is greater than or equal to chi(G). This is perhaps the most famous conjecture in Graph Theory and remains open even for graphs G with alpha(G) less than or equal to 2. Let W_5 denote the wheel on six vertices. We establish more evidence for Hadwiger\u27s conjecture by proving that h(G) is greater than or equal to chi(G) for all graphs G such that alpha(G) is less than or equal to 2 and G does not contain W_5 as an induced subgraph. Our second topic is related to Ramsey theory, a field that has intrigued those who study combinatorics for many decades. Computing the classical Ramsey numbers is a notoriously difficult problem, leaving many basic questions unanswered even after more than 80 years. We study Ramsey numbers under Gallai-colorings. A Gallai-coloring of a complete graph is an edge-coloring such that no triangle is colored with three distinct colors. Given a graph H and an integer k at least 1, the Gallai-Ramsey number, denoted GR_k(H), is the least positive integer n such that every Gallai-coloring of K_n with at most k colors contains a monochromatic copy of H. It turns out that GR_k(H) is more well-behaved than the classical Ramsey number R_k(H), though finding exact values of GR_k(H) is far from trivial. We show that for all k at least 3, GR_k(C_{2n+1}) = n2^k+1 where n is 4, 5, 6 or 7, and GR_k(C_{2n+1}) is at most (n ln n)2^k-(k+1)n+1 for all n at least 8, where C_{2n+1} denotes a cycle on 2n+1 vertices
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