2,023 research outputs found
Hamilton cycles in graphs and hypergraphs: an extremal perspective
As one of the most fundamental and well-known NP-complete problems, the
Hamilton cycle problem has been the subject of intensive research. Recent
developments in the area have highlighted the crucial role played by the
notions of expansion and quasi-randomness. These concepts and other recent
techniques have led to the solution of several long-standing problems in the
area. New aspects have also emerged, such as resilience, robustness and the
study of Hamilton cycles in hypergraphs. We survey these developments and
highlight open problems, with an emphasis on extremal and probabilistic
approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page
limits, this final version is slightly shorter than the previous arxiv
versio
Free nilpotent and -type Lie algebras. Combinatorial and orthogonal designs
The aim of our paper is to construct pseudo -type algebras from the
covering free nilpotent two-step Lie algebra as the quotient algebra by an
ideal. We propose an explicit algorithm of construction of such an ideal by
making use of a non-degenerate scalar product. Moreover, as a bypass result, we
recover the existence of a rational structure on pseudo -type algebras,
which implies the existence of lattices on the corresponding pseudo -type
Lie groups. Our approach substantially uses combinatorics and reveals the
interplay of pseudo -type algebras with combinatorial and orthogonal
designs. One of the key tools is the family of Hurwitz-Radon orthogonal
matrices
Unsolved Problems in Spectral Graph Theory
Spectral graph theory is a captivating area of graph theory that employs the
eigenvalues and eigenvectors of matrices associated with graphs to study them.
In this paper, we present a collection of topics in spectral graph theory,
covering a range of open problems and conjectures. Our focus is primarily on
the adjacency matrix of graphs, and for each topic, we provide a brief
historical overview.Comment: v3, 30 pages, 1 figure, include comments from Clive Elphick, Xiaofeng
Gu, William Linz, and Dragan Stevanovi\'c, respectively. Thanks! This paper
will be published in Operations Research Transaction
Packing spanning graphs from separable families
Let be a separable family of graphs. Then for all positive
constants and and for every sufficiently large integer ,
every sequence of graphs of order and maximum
degree at most such that packs into . This improves results of
B\"ottcher, Hladk\'y, Piguet, and Taraz when is the class of trees
and of Messuti, R\"odl, and Schacht in the case of a general separable family.
The result also implies approximate versions of the Oberwolfach problem and of
the Tree Packing Conjecture of Gy\'arf\'as (1976) for the case that all trees
have maximum degree at most . The proof uses the local resilience of
random graphs and a special multi-stage packing procedure
IST Austria Thesis
We present solutions to several problems originating from geometry and discrete mathematics: existence of equipartitions, maps without Tverberg multiple points, and inscribing quadrilaterals. Equivariant obstruction theory is the natural topological approach to these type of questions. However, for the specific problems we consider it had yielded only partial or no results. We get our results by complementing equivariant obstruction theory with other techniques from topology and geometry
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