117 research outputs found
Digraph extremal problems, hypergraph extremal problems, and the densities of graph structures
AbstractWe consider extremal problems ‘of Turán type’ for r-uniform ordered hypergraphs, where multiple oriented edges are permitted up to multiplicity q. With any such ‘(r, q)-graph’ Gn we associate an r-linear form whose maximum over the standard (n − 1)-simplex in Rn is called the (graph-) density g(Gn) of Gn. If ex(n, L) is the maximum number of oriented hyperedges in an n-vertex (r, q)-graph not containing a member of L, limn→∞ ex(n, L)/nr is called the extremal density of L. Motivated, in part, from results for ordinary graphs, digraphs, and multigraphs, we establish relations between these two notions
Monochromatic Clique Decompositions of Graphs
Let be a graph whose edges are coloured with colours, and be a -tuple of graphs. A monochromatic -decomposition of is a partition of the edge set of such that each
part is either a single edge or forms a monochromatic copy of in colour
, for some . Let be the smallest
number , such that, for every order- graph and every
-edge-colouring, there is a monochromatic -decomposition with at
most elements. Extending the previous results of Liu and Sousa
["Monochromatic -decompositions of graphs", Journal of Graph Theory},
76:89--100, 2014], we solve this problem when each graph in is a
clique and is sufficiently large.Comment: 14 pages; to appear in J Graph Theor
The approximate Loebl-Komlós-Sós conjecture I: The sparse decomposition
In a series of four papers we prove the following relaxation of the Loebl-Komlós-Sós conjecture: For every α > 0 there exists a number k0 such that for every k > k0, every n-vertex graph G with at least (1/2 + α)n vertices of degree at least (1 + α)k contains each tree T of order k as a subgraph. The method to prove our result follows a strategy similar to approaches that employ the Szemerédi regularity lemma: We decompose the graph G, find a suitable combinatorial structure inside the decomposition, and then embed the tree T into G using this structure. Since for sparse graphs G, the decomposition given by the regularity lemma is not helpful, we use a more general decomposition technique. We show that each graph can be decomposed into vertices of huge degree, regular pairs (in the sense of the regularity lemma), and two other objects each exhibiting certain expansion properties. In this paper, we introduce this novel decomposition technique. In the three follow-up papers, we find a suitable combinatorial structure inside the decomposition, which we then use for embedding the tree. © 2017 the authors
The critical window for the classical Ramsey-Tur\'an problem
The first application of Szemer\'edi's powerful regularity method was the
following celebrated Ramsey-Tur\'an result proved by Szemer\'edi in 1972: any
K_4-free graph on N vertices with independence number o(N) has at most (1/8 +
o(1)) N^2 edges. Four years later, Bollob\'as and Erd\H{o}s gave a surprising
geometric construction, utilizing the isoperimetric inequality for the high
dimensional sphere, of a K_4-free graph on N vertices with independence number
o(N) and (1/8 - o(1)) N^2 edges. Starting with Bollob\'as and Erd\H{o}s in
1976, several problems have been asked on estimating the minimum possible
independence number in the critical window, when the number of edges is about
N^2 / 8. These problems have received considerable attention and remained one
of the main open problems in this area. In this paper, we give nearly
best-possible bounds, solving the various open problems concerning this
critical window.Comment: 34 page
Socially optimal contribution rate and cap in a proportional (DC) pension system
In our model, the government operates a mandatory proportional (DC) pension system to substitute for the low life-cycle savings of the lower-paid myopic workers, while maintaining the incentives of the higher-paid far-sighted ones in contributing to the system. The introduction of an appropriate cap on pension contribution (or its base)—excluding the earnings above the cap from the contribution base—raises the optimal contribution rate, helping more the lower-paid myopic workers and reserving enough room for the saving of higher-paid far-sighted ones. The social welfare is almost independent of the cap in a relatively wide interval but the maximal welfare is higher than the capless welfare by 0.3–4.5 %.info:eu-repo/semantics/publishedVersio
The history of degenerate (bipartite) extremal graph problems
This paper is a survey on Extremal Graph Theory, primarily focusing on the
case when one of the excluded graphs is bipartite. On one hand we give an
introduction to this field and also describe many important results, methods,
problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version
of our survey presented in Erdos 100. In this version 2 only a citation was
complete
Monte-Carlo sampling of energy-constrained quantum superpositions in high-dimensional Hilbert spaces
Recent studies into the properties of quantum statistical ensembles in
high-dimensional Hilbert spaces have encountered difficulties associated with
the Monte-Carlo sampling of quantum superpositions constrained by the energy
expectation value. A straightforward Monte-Carlo routine would enclose the
energy constrained manifold within a larger manifold, which is easy to sample,
for example, a hypercube. The efficiency of such a sampling routine decreases
exponentially with the increase of the dimension of the Hilbert space, because
the volume of the enclosing manifold becomes exponentially larger than the
volume of the manifold of interest. The present paper explores the ways to
optimise the above routine by varying the shapes of the manifolds enclosing the
energy-constrained manifold. The resulting improvement in the sampling
efficiency is about a factor of five for a 14-dimensional Hilbert space. The
advantage of the above algorithm is that it does not compromise on the rigorous
statistical nature of the sampling outcome and hence can be used to test other
more sophisticated Monte-Carlo routines. The present attempts to optimise the
enclosing manifolds also bring insights into the geometrical properties of the
energy constrained manifold itself.Comment: 9 pages, 7 figures, accepted for publication in European Physical
Journal
Linkages, key sectors and structural change: some new perspectives
Recent exchanges in the literature on the identification and role of key sectors in national and regional economies have highlighted the difficulties of consensus regarding terminology, appropriate measurement as well as economic interpretation. In this paper, some new perspectives are advanced which provide a more comprehensive view of an economy and offer the potential for uncovering alternative perspectives about the role of linkages and multipliers in input-output and expanded social accounting systems. The analysis draws on some pioneering work by Miyazawa in the identification of internal and external multiplier effects. The theoretical techniques are illustrated by reference to a set of input-output tables for the Brazilian economy. The paper thus provides a more comprehensive view than the ones proposed by Baer, Fonseca, and Guilhoto (1987), Hewings, Fonseca, Guilhoto, and Sonis (1989) and the recent contributions of Clements and Rossi (1991, 1992) that draw on some earlier work of Cella (1984)
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