498 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Bipartite graphs with no K6 minor

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    This publication is available at Elsevier via https://doi.org/10.1016/j.jctb.2023.08.005 © 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC-BY license (http://creativecommons.org/licenses/by/4.0/).A theorem of Mader shows that every graph with average degree at least eight has a K6 minor, and this is false if we replace eight by any smaller constant. Replacing average degree by minimum degree seems to make little difference: we do not know whether all graphs with minimum degree at least seven have K6 minors, but minimum degree six is certainly not enough. For every ε > 0 there are arbitrarily large graphs with average degree at least 8 − ε and minimum degree at least six, with no K6 minor. But what if we restrict ourselves to bipartite graphs? The first statement remains true: for every ε > 0 there are arbitrarily large bipartite graphs with average degree at least 8 − ε and no K6 minor. But surprisingly, going to minimum degree now makes a significant difference. We will show that every bipartite graph with minimum degree at least six has a K6 minor. Indeed, it is enough that every vertex in the larger part of the bipartition has degree at least six.NSF DMS-EPSRC, DMS-2120644 || EPSRC, EP/V007327/1 || NSF, DMS-2154169 || AFOSR, A9550-19-1-0187 || NSERC, RGPIN-2020-03912

    Developing International Mindedness through the Arts in the International Baccalaureate (IB) Diploma Programme (DP): An International Survey Design Conducted across all Continents

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    One distinct purpose of international education is to develop greater international understanding and intercultural competences. For the International Baccalaureate, this translates into students developing international mindedness throughout its programmes and courses. However, international mindedness is not measured and the impact of the programmes on the development of international mindedness remains mainly anecdotal. Furthermore, in the Diploma Programme, the choice of Arts courses is optional and the value of an Arts education, or specifically the value of taking a Diploma Programme Arts course in developing international mindedness, is equally unclear. This study investigated the development of international mindedness in students who opted for a Diploma Programme Arts course versus those who did not. The study followed a repeated measures, comparative and mixed-methods research design using a survey tool for data collection. The survey consisted of a quantitative section based on existing surveys and a qualitative section with six open-ended questions. The quantitative data showed an increase in intercultural knowledge and behaviours, while no change in attitudes, and a decrease in values was identified for both student groups, Diploma Programme Arts and Non-Arts-students. Furthermore, there was an increase in intercultural communication skills particularly in Diploma Programme Arts-students. Qualitative data analysis revealed a spectrum of categories of responses. The qualitative data also identified themes in addition to those identified in International Baccalaureate documentation and literature. Recommendations include for the International Baccalaureate Organization to integrate some of the emerging themes in their documentations, for example themes relating to adaptability and interconnectedness, which may also provide an interesting focus for curriculum design. Furthermore, curriculum and programme design should place a greater focus on the development of attitudes and values in the Diploma Programme and a reconsideration of the optionality of the Arts in this context

    A machine learning approach to constructing Ramsey graphs leads to the Trahtenbrot-Zykov problem.

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    Attempts at approaching the well-known and difficult problem of constructing Ramsey graphs via machine learning lead to another difficult problem posed by Zykov in 1963 (now commonly referred to as the Trahtenbrot-Zykov problem): For which graphs F does there exist some graph G such that the neighborhood of every vertex in G induces a subgraph isomorphic to F? Chapter 1 provides a brief introduction to graph theory. Chapter 2 introduces Ramsey theory for graphs. Chapter 3 details a reinforcement learning implementation for Ramsey graph construction. The implementation is based on board game software, specifically the AlphaZero program and its success learning to play games from scratch. The chapter ends with a description of how computing challenges naturally shifted the project towards the Trahtenbrot-Zykov problem. Chapter 3 also includes recommendations for continuing the project and attempting to overcome these challenges. Chapter 4 defines the Trahtenbrot-Zykov problem and outlines its history, including proofs of results omitted from their original papers. This chapter also contains a program for constructing graphs with all neighborhood-induced subgraphs isomorphic to a given graph F. The end of Chapter 4 presents constructions from the program when F is a Ramsey graph. Constructing such graphs is a non-trivial task, as Bulitko proved in 1973 that the Trahtenbrot-Zykov problem is undecidable. Chapter 5 is a translation from Russian to English of this famous result, a proof not previously available in English. Chapter 6 introduces Cayley graphs and their relationship to the Trahtenbrot-Zykov problem. The chapter ends with constructions of Cayley graphs Γ in which the neighborhood of every vertex of Γ induces a subgraph isomorphic to a given Ramsey graph, which leads to a conjecture regarding the unique extremal Ramsey(4, 4) graph

    Planar Disjoint Paths, Treewidth, and Kernels

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    In the Planar Disjoint Paths problem, one is given an undirected planar graph with a set of kk vertex pairs (si,ti)(s_i,t_i) and the task is to find kk pairwise vertex-disjoint paths such that the ii-th path connects sis_i to tit_i. We study the problem through the lens of kernelization, aiming at efficiently reducing the input size in terms of a parameter. We show that Planar Disjoint Paths does not admit a polynomial kernel when parameterized by kk unless coNP \subseteq NP/poly, resolving an open problem by [Bodlaender, Thomass{\'e}, Yeo, ESA'09]. Moreover, we rule out the existence of a polynomial Turing kernel unless the WK-hierarchy collapses. Our reduction carries over to the setting of edge-disjoint paths, where the kernelization status remained open even in general graphs. On the positive side, we present a polynomial kernel for Planar Disjoint Paths parameterized by k+twk + tw, where twtw denotes the treewidth of the input graph. As a consequence of both our results, we rule out the possibility of a polynomial-time (Turing) treewidth reduction to tw=kO(1)tw= k^{O(1)} under the same assumptions. To the best of our knowledge, this is the first hardness result of this kind. Finally, combining our kernel with the known techniques [Adler, Kolliopoulos, Krause, Lokshtanov, Saurabh, Thilikos, JCTB'17; Schrijver, SICOMP'94] yields an alternative (and arguably simpler) proof that Planar Disjoint Paths can be solved in time 2O(k2)nO(1)2^{O(k^2)}\cdot n^{O(1)}, matching the result of [Lokshtanov, Misra, Pilipczuk, Saurabh, Zehavi, STOC'20].Comment: To appear at FOCS'23, 82 pages, 30 figure

    Computation and Physics in Algebraic Geometry

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    Physics provides new, tantalizing problems that we solve by developing and implementing innovative and effective geometric tools in nonlinear algebra. The techniques we employ also rely on numerical and symbolic computations performed with computer algebra. First, we study solutions to the Kadomtsev-Petviashvili equation that arise from singular curves. The Kadomtsev-Petviashvili equation is a partial differential equation describing nonlinear wave motion whose solutions can be built from an algebraic curve. Such a surprising connection established by Krichever and Shiota also led to an entirely new point of view on a classical problem in algebraic geometry known as the Schottky problem. To explore the connection with curves with at worst nodal singularities, we define the Hirota variety, which parameterizes KP solutions arising from such curves. Studying the geometry of the Hirota variety provides a new approach to the Schottky problem. We investigate it for irreducible rational nodal curves, giving a partial solution to the weak Schottky problem in this case. Second, we formulate questions from scattering amplitudes in a broader context using very affine varieties and D-module theory. The interplay between geometry and combinatorics in particle physics indeed suggests an underlying, coherent mathematical structure behind the study of particle interactions. In this thesis, we gain a better understanding of mathematical objects, such as moduli spaces of point configurations and generalized Euler integrals, for which particle physics provides concrete, non-trivial examples, and we prove some conjectures stated in the physics literature. Finally, we study linear spaces of symmetric matrices, addressing questions motivated by algebraic statistics, optimization, and enumerative geometry. This includes giving explicit formulas for the maximum likelihood degree and studying tangency problems for quadric surfaces in projective space from the point of view of real algebraic geometry

    Parameterized Graph Modification Beyond the Natural Parameter

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    Higher-Point Integrands in N=4 super Yang-Mills Theory

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    We compute the integrands of five-, six-, and seven-point correlation functions of twenty-prime operators with general polarizations at the two-loop order in N=4 super Yang-Mills theory. In addition, we compute the integrand of the five-point function at three-loop order. Using the operator product expansion, we extract the two-loop four-point function of one Konishi operator and three twenty-prime operators. Two methods were used for computing the integrands. The first method is based on constructing an ansatz, and then numerically fitting for the coefficients using the twistor-space reformulation of N=4 super Yang-Mills theory. The second method is based on the OPE decomposition. Only very few correlator integrands for more than four points were known before. Our results can be used to test conjectures, and to make progresses on the integrability-based hexagonalization approach for correlation functions.Comment: 50 pages, 2 figures. v2: many changes and additions, files added, references update

    On the choosability of HH-minor-free graphs

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    Given a graph HH, let us denote by fχ(H)f_\chi(H) and f(H)f_\ell(H), respectively, the maximum chromatic number and the maximum list chromatic number of HH-minor-free graphs. Hadwiger's famous coloring conjecture from 1943 states that fχ(Kt)=t1f_\chi(K_t)=t-1 for every t2t \ge 2. In contrast, for list coloring it is known that 2to(t)f(Kt)O(t(loglogt)6)2t-o(t) \le f_\ell(K_t) \le O(t (\log \log t)^6) and thus, f(Kt)f_\ell(K_t) is bounded away from the conjectured value t1t-1 for fχ(Kt)f_\chi(K_t) by at least a constant factor. The so-called HH-Hadwiger's conjecture, proposed by Seymour, asks to prove that fχ(H)=v(H)1f_\chi(H)=\textsf{v}(H)-1 for a given graph HH (which would be implied by Hadwiger's conjecture). In this paper, we prove several new lower bounds on f(H)f_\ell(H), thus exploring the limits of a list coloring extension of HH-Hadwiger's conjecture. Our main results are: For every ε>0\varepsilon>0 and all sufficiently large graphs HH we have f(H)(1ε)(v(H)+κ(H))f_\ell(H)\ge (1-\varepsilon)(\textsf{v}(H)+\kappa(H)), where κ(H)\kappa(H) denotes the vertex-connectivity of HH. For every ε>0\varepsilon>0 there exists C=C(ε)>0C=C(\varepsilon)>0 such that asymptotically almost every nn-vertex graph HH with Cnlogn\left\lceil C n\log n\right\rceil edges satisfies f(H)(2ε)nf_\ell(H)\ge (2-\varepsilon)n. The first result generalizes recent results on complete and complete bipartite graphs and shows that the list chromatic number of HH-minor-free graphs is separated from the natural lower bound (v(H)1)(\textsf{v}(H)-1) by a constant factor for all large graphs HH of linear connectivity. The second result tells us that even when HH is a very sparse graph (with an average degree just logarithmic in its order), f(H)f_\ell(H) can still be separated from (v(H)1)(\textsf{v}(H)-1) by a constant factor arbitrarily close to 22. Conceptually these results indicate that the graphs HH for which f(H)f_\ell(H) is close to (v(H)1)(\textsf{v}(H)-1) are typically rather sparse.Comment: 14 page

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum
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