18 research outputs found
The number of Hamiltonian decompositions of regular graphs
A Hamilton cycle in a graph is a cycle passing through every vertex
of . A Hamiltonian decomposition of is a partition of its edge
set into disjoint Hamilton cycles. One of the oldest results in graph theory is
Walecki's theorem from the 19th century, showing that a complete graph on
an odd number of vertices has a Hamiltonian decomposition. This result was
recently greatly extended by K\"{u}hn and Osthus. They proved that every
-regular -vertex graph with even degree for some fixed
has a Hamiltonian decomposition, provided is sufficiently
large. In this paper we address the natural question of estimating ,
the number of such decompositions of . Our main result is that
. In particular, the number of Hamiltonian
decompositions of is
Rainbow Hamilton cycles in random regular graphs
A rainbow subgraph of an edge-coloured graph has all edges of distinct
colours. A random d-regular graph with d even, and having edges coloured
randomly with d/2 of each of n colours, has a rainbow Hamilton cycle with
probability tending to 1 as n tends to infinity, provided d is at least 8.Comment: 16 page
Uncoverings on graphs and network reliability
We propose a network protocol similar to the -tree protocol of Itai and
Rodeh [{\em Inform.\ and Comput.}\ {\bf 79} (1988), 43--59]. To do this, we
define an {\em -uncovering-by-bases} for a connected graph to be a
collection of spanning trees for such that any -subset of
edges of is disjoint from at least one tree in , where is
some integer strictly less than the edge connectivity of . We construct
examples of these for some infinite families of graphs. Many of these infinite
families utilise factorisations or decompositions of graphs. In every case the
size of the uncovering-by-bases is no larger than the number of edges in the
graph and we conjecture that this may be true in general.Comment: 12 pages, 5 figure
Вклад теплового расширения решетки в температурные изменения ширины запрещенной зоны полупроводника CuGa5Se8
KUSCHNER T.L. СHUGUNOV S.V. Contributions of thermal expansion on the changes in temperature of the optical band gap in semiconductor
CuGa5Se8Зависимости ширины запрещенной зоны от температуры для
монокристаллов CuGa5Se8 были проанализированы с помощью
модели, которая учитывает как электрон-фононное взаимодействие,
так и тепловое расширение кристаллической решетки. Обнаружено,
что на температурные изменения ширины запрещенной зоны влияют в основном оптические фононы с эффективной энергией равной
приблизительно 16 мэВ. Из значений параметра Θ была рассчитана
температура Дебая, значение которой хорошо согласуется с данными, полученными ранее из температурных рентгеновских измерений
Backtracking algorithms for constructing the Hamiltonian decomposition of a 4-regular multigraph
We consider a Hamiltonian decomposition problem of partitioning a regular
multigraph into edge-disjoint Hamiltonian cycles. It is known that verifying
vertex nonadjacency in the 1-skeleton of the symmetric and asymmetric traveling
salesperson polytopes is an NP-complete problem. On the other hand, a
sufficient condition for two vertices to be nonadjacent can be formulated as a
combinatorial problem of finding a Hamiltonian decomposition of a 4-regular
multigraph. We present two backtracking algorithms for verifying vertex
nonadjacency in the 1-skeleton of the traveling salesperson polytope and
constructing a Hamiltonian decomposition: an algorithm based on a simple path
extension and an algorithm based on the chain edge fixing procedure. According
to the results of computational experiments for undirected multigraphs, both
backtracking algorithms lost to the known general variable neighborhood search
algorithm. However, for directed multigraphs, the algorithm based on chain edge
fixing showed comparable results with heuristics on instances with the existing
solution and better results on instances of the problem where the Hamiltonian
decomposition does not exist.Comment: In Russian. Computational experiments are revise
Алгоритмы поиска с возвратом для построения гамильтонова разложения 4-регулярного мультиграфа
We consider a Hamiltonian decomposition problem of partitioning a regular graph into edge-disjoint Hamiltonian cycles. It is known that verifying vertex non-adjacency in the 1-skeleton of the symmetric and asymmetric traveling salesperson polytopes is an NP-complete problem. On the other hand, a suffcient condition for two vertices to be non-adjacent can be formulated as a combinatorial problem of finding a Hamiltonian decomposition of a 4-regular multigraph. We present two backtracking algorithms for verifying vertex non-adjacency in the 1-skeleton of the traveling salesperson polytope and constructing a Hamiltonian decomposition: an algorithm based on a simple path extension and an algorithm based on the chain edge fixing procedure. Based on the results of the computational experiments for undirected multigraphs, both backtracking algorithms lost to the known heuristic general variable neighborhood search algorithm. However, for directed multigraphs, the algorithm based on chain fixing of edges showed comparable results with heuristics on instances with existing solutions, and better results on instances of the problem where the Hamiltonian decomposition does not exist.Рассматривается задача построения гамильтонова разложения регулярного мультиграфа на гамильтоновы циклы без общих рёбер. Известно, что проверка несмежности вершин в полиэдральных графах симметричного и асимметричного многогранников коммивояжёра является NP-полной задачей. С другой стороны, достаточное условие несмежности вершин можно сформулировать в виде комбинаторной задачи построения гамильтонова разложения 4-регулярного мультиграфа. В статье представлены два алгоритма поиска с возвратом для проверки несмежности вершин в полиэдральном графе коммивояжёра и построения гамильтонова разложения 4-регулярного мультиграфа: алгоритм на основе последовательного расширения простого пути и алгоритм на основе процедуры цепного фиксирования рёбер. По результатам вычислительных экспериментов для неориентированных мультиграфов оба переборных алгоритма проиграли известному эвристическому алгоритму поиска с переменными окрестностями. Однако для ориентированных мультиграфов алгоритм на основе цепного фиксирования рёбер показал сопоставимые результаты с эвристиками на экземплярах задачи, имеющих решение, и лучшие результаты на экземплярах задачи, для которых гамильтонова разложения не существует
Achieving the Optimal Steaming Capacity and Delay Using Random Regular Digraphs in P2P Networks
In earlier work, we showed that it is possible to achieve
streaming delay with high probability in a peer-to-peer network, where each
peer has as little as four neighbors, while achieving any arbitrary fraction of
the maximum possible streaming rate. However, the constant in the
delay term becomes rather large as we get closer to the maximum streaming rate.
In this paper, we design an alternative pairing and chunk dissemination
algorithm that allows us to transmit at the maximum streaming rate while
ensuring that all, but a negligible fraction of the peers, receive the data
stream with delay with high probability. The result is established
by examining the properties of graph formed by the union of two or more random
1-regular digraphs, i.e., directed graphs in which each node has an incoming
and an outgoing node degree both equal to one
Edge-colorings of 4-regular graphs with the minimum number of palettes
A proper edge-coloring of a graph G is an assignment of colors to the edges of G such that adjacent edges receive distinct
colors. A proper edge-coloring defines at each vertex the set of colors of its incident edges. Following the terminology introduced by Hor\u148\ue1k, Kalinowski, Meszka and Wo\u17aniak, we call such a set of colors the palette of the vertex. What is the minimum number of distinct palettes taken over all proper edge-colorings of G? A complete answer is known for complete graphs and cubic graphs. We study in some detail the problem for 4-regular graphs
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
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