scholarly journals Hamilton Paths and Cycles in Varietal Hypercube Networks with Mixed Faults

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Jian-Guang Zhou ◽  
Jun-Ming Xu
Keyword(s):  
Hamilton Cycle ◽  
Hamilton Path ◽  
Inductive Construction ◽  
Hypercube Network ◽  
Hypercube Networks ◽  
Hamilton Paths

This paper considers the varietal hypercube network VQn with mixed faults and shows that VQn contains a fault-free Hamilton cycle provided faults do not exceed n-2 for n⩾2 and contains a fault-free Hamilton path between any pair of vertices provided faults do not exceed n-3 for n⩾3. The proof is based on an inductive construction.

scholarly journals Spanning Trees of Bounded Degree

10.37236/1577 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Andrzej Czygrinow ◽  
Genghua Fan ◽  
Glenn Hurlbert ◽  
H. A. Kierstead ◽  
William T. Trotter
Keyword(s):  
Spanning Tree ◽  
Maximum Degree ◽  
Additional Assumption ◽  
Spanning Trees ◽  
Independent Set ◽  
Hamilton Cycle ◽  
Bounded Degree ◽  
Hamilton Path ◽  
Classic Theorem

Dirac's classic theorem asserts that if ${\bf G}$ is a graph on $n$ vertices, and $\delta({\bf G})\ge n/2$, then ${\bf G}$ has a hamilton cycle. As is well known, the proof also shows that if $\deg(x)+\deg(y)\ge(n-1)$, for every pair $x$, $y$ of independent vertices in ${\bf G}$, then ${\bf G}$ has a hamilton path. More generally, S. Win has shown that if $k\ge 2$, ${\bf G}$ is connected and $\sum_{x\in I}\deg(x)\ge n-1$ whenever $I$ is a $k$-element independent set, then ${\bf G}$ has a spanning tree ${\bf T}$ with $\Delta({\bf T})\le k$. Here we are interested in the structure of spanning trees under the additional assumption that ${\bf G}$ does not have a spanning tree with maximum degree less than $k$. We show that apart from a single exceptional class of graphs, if $\sum_{x\in I}\deg(x)\ge n-1$ for every $k$-element independent set, then ${\bf G}$ has a spanning caterpillar ${\bf T}$ with maximum degree $k$. Furthermore, given a maximum path $P$ in ${\bf G}$, we may require that $P$ is the spine of ${\bf T}$ and that the set of all vertices whose degree in ${\bf T}$ is $3$ or larger is independent in ${\bf T}$.

scholarly journals Hypercube Network Fault Tolerance: A Probabilistic Approach

2005 ◽  
Vol 06 (01) ◽  
pp. 17-34 ◽  
Author(s):  
JIANER CHEN ◽  
IYAD A. KANJ ◽  
GUOJUN WANG
Keyword(s):  
Fault Tolerance ◽  
Lower Bounds ◽  
Network Connectivity ◽  
Point Of View ◽  
Theoretical Point ◽  
Node Failure ◽  
Hypercube Network ◽  
Network Fault Tolerance ◽  
Hypercube Networks ◽  
Network Fault

Extensive experiments and experience have shown that the well-known hypercube networks are highly fault tolerant. What is frustrating is that it seems very difficult to properly formulate and formally prove this important fact, despite extensive research efforts in the past two decades. Most proposed fault tolerance models for hypercube networks are only able to characterize very rare extreme situations thus significantly underestimating the fault tolerance power of hypercube networks, while for more realistic fault tolerance models, the analysis becomes much more complicated. In this paper, we develop new techniques that enable us to analyze a more realistic fault tolerance model and derive lower bounds for the probability of hypercube network fault tolerance in terms of node failure probability. Our results are both theoretically significant and practically important. From the theoretical point of view, our method offers very general and powerful techniques for formally proving lower bounds on the probability of network connectivity, while from the practical point of view, our results provide formally proven and precisely given upper bounds on node failure probabilities for manufacturers to achieve a desired probability for network connectivity. Our techniques are also useful and powerful for analysis of the performance of routing algorithms, and applicable to the study of other hierarchical network structures and to other network communication problems.

A Hamilton Path Embedding Algorithm on DCell

2013 ◽  
Vol 336-338 ◽  
pp. 2468-2471
Author(s):  
Xi Wang ◽  
Jian Xi Fan ◽  
Bao Lei Cheng ◽  
Wen Jun Liu ◽  
Fei Fei Li
Keyword(s):  
Fault Tolerance ◽  
Time Complexity ◽  
Data Centers ◽  
Network Capacity ◽  
Node Number ◽  
Hamilton Path ◽  
Hamilton Paths

DCell has been proposed for enormous data centers as a server centric interconnection which can support millions of servers with high network capacity and provide good fault tolerance by only using commodity switches. In this paper, we study an attractive algorithm of finding Hamilton paths in dimensional. The time complexity of this algorithm is , where is the node number of DCellk .

Hamilton cycle and Hamilton path extendability of Cayley graphs on abelian groups

2011 ◽  
Vol 70 (4) ◽  
pp. 384-403 ◽  
Author(s):  
Štefko MiklaviČ ◽  
Primož Šparl
Keyword(s):  
Abelian Groups ◽  
Cayley Graphs ◽  
Hamilton Cycle ◽  
Hamilton Path

scholarly journals Transition Restricted Gray Codes

10.37236/1235 ◽  
1996 ◽  
Vol 3 (1) ◽  
Author(s):  
Bette Bultena ◽  
Frank Ruskey
Keyword(s):  
Undirected Graph ◽  
Gray Code ◽  
Hamilton Cycle ◽  
Gray Codes ◽  
Hamilton Path ◽  
Vertex Set ◽  
G Code

A Gray code is a Hamilton path $H$ on the $n$-cube, $Q_n$. By labeling each edge of $Q_n$ with the dimension that changes between its incident vertices, a Gray code can be thought of as a sequence $H = t_1,t_2,\ldots,t_{N-1}$ (with $N = 2^n$ and each $t_i$ satisfying $1 \le t_i \le n$). The sequence $H$ defines an (undirected) graph of transitions, $G_H$, whose vertex set is $\{1,2,\ldots,n\}$ and whose edge set $E(G_H) = \{ [t_i,t_{i+1}] \mid 1 \le i \le N-1 \}$. A $G$-code is a Hamilton path $H$ whose graph of transitions is a subgraph of $G$; if $H$ is a Hamilton cycle then it is a cyclic $G$-code. The classic binary reflected Gray code is a cyclic $K_{1,n}$-code. We prove that every tree $T$ of diameter 4 has a $T$-code, and that no tree $T$ of diameter 3 has a $T$-code.

scholarly journals On the Maximum Number of Spanning Copies of an Orientation in a Tournament

2017 ◽  
Vol 26 (5) ◽  
pp. 775-796
Author(s):  
RAPHAEL YUSTER
Keyword(s):  
Lower Bound ◽  
Hamilton Cycle ◽  
Constant Factor ◽  
Expected Number ◽  
Bounded Degree ◽  
Hamilton Path ◽  
Random Tournament ◽  
Regular Tournament

For an orientation H with n vertices, let T(H) denote the maximum possible number of labelled copies of H in an n-vertex tournament. It is easily seen that T(H) ≥ n!/2e(H), as the latter is the expected number of such copies in a random tournament. For n odd, let R(H) denote the maximum possible number of labelled copies of H in an n-vertex regular tournament. In fact, Adler, Alon and Ross proved that for H=Cn, the directed Hamilton cycle, T(Cn) ≥ (e−o(1))n!/2n, and it was observed by Alon that already R(Cn) ≥ (e−o(1))n!/2n. Similar results hold for the directed Hamilton path Pn. In other words, for the Hamilton path and cycle, the lower bound derived from the expectation argument can be improved by a constant factor. In this paper we significantly extend these results, and prove that they hold for a larger family of orientations H which includes all bounded-degree Eulerian orientations and all bounded-degree balanced orientations, as well as many others. One corollary of our method is that for any fixed k, every k-regular orientation H with n vertices satisfies T(H) ≥ (ek−o(1))n!/2e(H), and in fact, for n odd, R(H) ≥ (ek−o(1))n!/2e(H).

ROUTING IN HYPERCUBE NETWORKS WITH A CONSTANT FRACTION OF FAULTY NODES

2001 ◽  
Vol 02 (03) ◽  
pp. 283-294 ◽  
Author(s):  
JIANER CHEN ◽  
GUOJUN WANG ◽  
SONGQIAO CHEN
Keyword(s):  
Hamming Distance ◽  
Linear Time ◽  
Local Information ◽  
Routing Algorithms ◽  
Global Information ◽  
Natural Conditions ◽  
Hypercube Network ◽  
Hypercube Networks

We consider routing in hypercube networks with a very large number (i.e., up to a constant fraction) of faulty nodes. Simple and natural conditions are identified under which hypercube networks with a very large number of faulty nodes still remain connected. The conditions can be detected and maintained in a distributed manner based on localized management. Efficient routing algorithms on hypercube networks satisfying these conditions are developed. For a hypercube network that satisfies the conditions and may contain up of 37.5% faulty nodes, our algorithms run in linear time and for any two given non-faulty nodes find a routing path of length bounded by four times the Hamming distance between the two nodes. Moreover, our algorithms are distributed and local-information-based in the sense that each node in the network knows only its neighbors' status and no global information of the network is required by the algorithms.

scholarly journals Bent Hamilton Cycles in $d$-Dimensional Grid Graphs

10.37236/1694 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
F. Ruskey ◽  
Joe Sawada
Keyword(s):  
Hamilton Cycle ◽  
Graph Product ◽  
Grid Graph ◽  
Hamilton Cycles ◽  
Hamilton Path ◽  
Successive Pair ◽  
Grid Graphs

A bent Hamilton cycle in a grid graph is one in which each edge in a successive pair of edges lies in a different dimension. We show that the $d$-dimensional grid graph has a bent Hamilton cycle if some dimension is even and $d \geq 3$, and does not have a bent Hamilton cycle if all dimensions are odd. In the latter case, we determine the conditions for when a bent Hamilton path exists. For the $d$-dimensional toroidal grid graph (i.e., the graph product of $d$ cycles), we show that there exists a bent Hamilton cycle when all dimensions are odd and $d \geq 3$. We also show that if $d=2$, then there exists a bent Hamilton cycle if and only if both dimensions are even.

scholarly journals Counting Hamilton cycles in Dirac hypergraphs

2020 ◽  
pp. 1-23
Author(s):  
Stefan Glock ◽  
Stephen Gould ◽  
Felix Joos ◽  
Daniela Kühn ◽  
Deryk Osthus
Keyword(s):  
Hamilton Cycle ◽  
Uniform Hypergraph ◽  
Similar Estimate ◽  
Hamilton Cycles

Abstract A tight Hamilton cycle in a k-uniform hypergraph (k-graph) G is a cyclic ordering of the vertices of G such that every set of k consecutive vertices in the ordering forms an edge. Rödl, Ruciński and Szemerédi proved that for $k\ge 3$ , every k-graph on n vertices with minimum codegree at least $n/2+o(n)$ contains a tight Hamilton cycle. We show that the number of tight Hamilton cycles in such k-graphs is ${\exp(n\ln n-\Theta(n))}$ . As a corollary, we obtain a similar estimate on the number of Hamilton ${\ell}$ -cycles in such k-graphs for all ${\ell\in\{0,\ldots,k-1\}}$ , which makes progress on a question of Ferber, Krivelevich and Sudakov.

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