498 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Bipartite graphs with no K6 minor
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
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.
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
In the Planar Disjoint Paths problem, one is given an undirected planar graph
with a set of vertex pairs and the task is to find pairwise
vertex-disjoint paths such that the -th path connects to . 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 unless coNP
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 , where 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 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 , 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
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
Higher-Point Integrands in N=4 super Yang-Mills Theory
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 -minor-free graphs
Given a graph , let us denote by and ,
respectively, the maximum chromatic number and the maximum list chromatic
number of -minor-free graphs. Hadwiger's famous coloring conjecture from
1943 states that for every . In contrast, for list
coloring it is known that
and thus, is bounded away from the conjectured value for
by at least a constant factor. The so-called -Hadwiger's
conjecture, proposed by Seymour, asks to prove that
for a given graph (which would be implied by Hadwiger's conjecture). In
this paper, we prove several new lower bounds on , thus exploring
the limits of a list coloring extension of -Hadwiger's conjecture. Our main
results are:
For every and all sufficiently large graphs we have
, where
denotes the vertex-connectivity of .
For every there exists such that
asymptotically almost every -vertex graph with edges satisfies .
The first result generalizes recent results on complete and complete
bipartite graphs and shows that the list chromatic number of -minor-free
graphs is separated from the natural lower bound by a
constant factor for all large graphs of linear connectivity. The second
result tells us that even when is a very sparse graph (with an average
degree just logarithmic in its order), can still be separated from
by a constant factor arbitrarily close to . Conceptually
these results indicate that the graphs for which is close to
are typically rather sparse.Comment: 14 page
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
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