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A Packet Routing Protocol for Arbitrary Networks
 In Proceedings of the 12th Symposium on Theoretical Aspects of Computer Science
, 1995
"... . In this paper, we introduce an online protocol which routes any set of packets along shortest paths through an arbitrary Nnode network in O(congestion + diameter + log N) rounds, with high probability. This time bound is optimal up to the additive log N , and it was previously only reached for ..."
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Cited by 31 (16 self)
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. In this paper, we introduce an online protocol which routes any set of packets along shortest paths through an arbitrary Nnode network in O(congestion + diameter + log N) rounds, with high probability. This time bound is optimal up to the additive log N , and it was previously only reached for boundeddegree levelled networks. Further, we prove bounds on the congestion of random routing problems for Cayley networks and general node symmetric networks based on the construction of shortest paths systems. In particular, we give construction schemes for shortest paths systems and show that if every processor sends p packets to random destinations along the paths described in the paths system, then the congestion is bounded by O(p \Delta diameter + log N ), with high probability. Finally, we prove an (apparently suboptimal) congestion bound for random routing problems on randomly chosen regular networks. 1 Introduction Communication among the processors of a parallel computer usually ...
Smarandache MultiSpace Theory
, 2011
"... Our WORLD is a multiple one both shown by the natural world and human beings. For example, the observation enables one knowing that there are infinite planets in the universe. Each of them revolves on its own axis and has its own seasons. In the human society, these rich or poor, big or small countr ..."
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Cited by 13 (5 self)
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Our WORLD is a multiple one both shown by the natural world and human beings. For example, the observation enables one knowing that there are infinite planets in the universe. Each of them revolves on its own axis and has its own seasons. In the human society, these rich or poor, big or small countries appear and each of them has its own system. All of these show that our WORLD is not in homogenous but in multiple. Besides, all things that one can acknowledge is determined by his eyes, or ears, or nose, or tongue, or body or passions, i.e., these six organs, which means the WORLD consists of have and not have parts for human beings. For thousands years, human being has never stopped his steps for exploring its behaviors of all kinds. We are used to the idea that our space has three dimensions: length, breadth and height with time providing the fourth dimension of spacetime by Einstein. In the string or superstring theories, we encounter 10 dimensions. However, we do not even know what the right degree of freedom is, as Witten said. Today, we have known two heartening notions for sciences. One is the Smarandache multispace came into being by purely logic.
Shortest paths routing in arbitrary networks
 JOURNAL OF ALGORITHMS
, 1999
"... We introduce an online protocol which routes any set of N packets along shortest paths with congestion C and dilation D through an arbitrary network in OC � Ž D � log N. steps, with high probability. This time bound is optimal up to the additive log N, and it has previously only been reached for bo ..."
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Cited by 11 (2 self)
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We introduce an online protocol which routes any set of N packets along shortest paths with congestion C and dilation D through an arbitrary network in OC � Ž D � log N. steps, with high probability. This time bound is optimal up to the additive log N, and it has previously only been reached for boundeddegree leveled networks. Further, we show that the preceding bound holds also for random routing problems with C denoting the maximum expected congestion over all links. Based on this result, we give applications for random routing in Cayley networks, general node symmetric networks, edge symmetric networks, and de Bruijn networks. Finally, we examine the problems arising when our approach is applied to routing along nonshortest paths, deterministic routing, or routing with bounded buffers.
Chromatic Index Critical Graphs of Even Order with Five Major Vertices
"... We prove that there does not exist any chromatic index critical graph of even order with exactly five vertices of maximum degree. This extends an earlier result of Chetwynd and Hilton who proved the same with five replaced by four or three. 1 ..."
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Cited by 5 (0 self)
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We prove that there does not exist any chromatic index critical graph of even order with exactly five vertices of maximum degree. This extends an earlier result of Chetwynd and Hilton who proved the same with five replaced by four or three. 1
Chromatic Index Critical Graphs of Orders 11 and 12
, 1997
"... A chromaticindexcritical graph G on n vertices is nontrivial if it has at most \Deltab n 2 c edges. We prove that there is no chromaticindexcritical graph of order 12, and that there are precisely two nontrivial chromatic index critical graphs on 11 vertices. Together with known results thi ..."
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Cited by 3 (1 self)
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A chromaticindexcritical graph G on n vertices is nontrivial if it has at most \Deltab n 2 c edges. We prove that there is no chromaticindexcritical graph of order 12, and that there are precisely two nontrivial chromatic index critical graphs on 11 vertices. Together with known results this implies that there are precisely three nontrivial chromaticindex critical graphs of order 12. 1 Introduction A famous theorem of Vizing [20] states that the chromatic index Ø 0 (G) of a simple graph G is \Delta(G) or \Delta(G) + 1, where \Delta(G) denotes the maximum vertex degree in G. A graph G is class 1 if Ø 0 (G) = \Delta(G) and it is class 2 otherwise. A class 2 graph G is (chromatic index) critical if Ø 0 (G \Gamma e) ! Ø 0 (G) for each edge e of G. If we want to stress the maximum vertex degree of a critical graph G we say G is \Delta(G)critical. Critical graphs of odd order are easy to construct while not much is known about critical graphs of even order. One reas...
Universal StoreandForward Routing
, 1996
"... We introduce an online protocol which routes any set of N packets along shortest paths with congestion C through an arbitrary network G in O(C + diam(G) + log N) steps, with high probability. This time bound is optimal up to the additive log N , and it has previously only been reached for bounded ..."
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Cited by 1 (1 self)
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We introduce an online protocol which routes any set of N packets along shortest paths with congestion C through an arbitrary network G in O(C + diam(G) + log N) steps, with high probability. This time bound is optimal up to the additive log N , and it has previously only been reached for boundeddegree leveled networks. Further, we show that the above bound holds also for random routing problems with C denoting the maximum expected congestion over all links. Based on this result, we give applications for random routing in Cayley networks, general node symmetric networks, edge symmetric networks, and de Bruijn networks. Finally, we examine the problems arising when we try to apply our approach to routing along nonshortest paths, deterministic routing, or routing with bounded buffers. 1 Introduction Communication among the processors of a parallel computer usually requires a large portion of runtime of a parallel algorithm. These computers are often realized as relatively sparse net...
ChromaticIndex Critical Graphs of Even Order
, 1997
"... A kcritrical graph G has maximum degree k 0, chromatic index Ø 0 (G) = k + 1 and Ø 0 (G \Gamma e) ! k + 1 for each edge e of G. The Critical Graph Conjecture, Jakobsen [8] and Beineke, Wilson [1], claims that every kcritical graph is of odd order. Fiorini and Wilson [6] conjectured that ev ..."
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A kcritrical graph G has maximum degree k 0, chromatic index Ø 0 (G) = k + 1 and Ø 0 (G \Gamma e) ! k + 1 for each edge e of G. The Critical Graph Conjecture, Jakobsen [8] and Beineke, Wilson [1], claims that every kcritical graph is of odd order. Fiorini and Wilson [6] conjectured that every kcritical graph of even order has a 1factor. Chetwynd and Yap [4] stated the problem whether it is true that if G is a kcritical graph of odd order, then G \Gamma v has a 1factor for every vertex v of minimum degree. These conjectures are disproved and the problem is answered in the negative for k 2 f3; 4g. We disprove these conjectures and answer the problem in the negative for all k 3. We also construct kcritical graphs on n vertices with degree sequence 23 2 4 n\Gamma3 , answering a question of Yap [11]. 1 Introduction We consider connected multigraphs M = (V (M); E(M)) without loops, where V (M) (E(M)) denotes the set of vertices (edges) of M . The degree dM (v) of a v...
Smarandache multispace theory (IV)  Applications to theoretical physics
, 2006
"... A Smarandache multispace is a union of n different spaces equipped with some different structures for an integer n ≥ 2, which can be both used for discrete or connected spaces, particularly for geometries and spacetimes in theoretical physics. This monograph concentrates on characterizing various ..."
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A Smarandache multispace is a union of n different spaces equipped with some different structures for an integer n ≥ 2, which can be both used for discrete or connected spaces, particularly for geometries and spacetimes in theoretical physics. This monograph concentrates on characterizing various multispaces including three parts altogether. The first part is on algebraic multispaces with structures, such as those of multigroups, multirings, multivector spaces, multimetric spaces, multioperation systems and multimanifolds, also multivoltage graphs, multiembedding of a graph in an nmanifold, · · ·, etc.. The second discusses Smarandache geometries, including those of map geometries, planar map geometries and pseudoplane geometries, in which the Finsler geometry, particularly the Riemann geometry appears as a special case of these Smarandache geometries. The third part of this book considers the applications of multispaces to theoretical physics, including the relativity theory, the Mtheory and the cosmology. Multispace models for pbranes and cosmos are constructed and some questions in cosmology are clarified by multispaces. The first two parts are relative independence for reading and in each part open problems are included for further research of interested readers.
Gunnar Brinkmann
, 2003
"... A graph is chromaticindexcritical if it cannot be edgecoloured with ∆ colours (with ∆ the maximal degree of the graph), and if the removal of any edge decreases its chromatic index. The Critical Graph Conjecture stated that any such graph has odd order. It has been proved false and the smallest k ..."
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A graph is chromaticindexcritical if it cannot be edgecoloured with ∆ colours (with ∆ the maximal degree of the graph), and if the removal of any edge decreases its chromatic index. The Critical Graph Conjecture stated that any such graph has odd order. It has been proved false and the smallest known counterexample has order 18 [18, 31]. In this paper we show that there are no chromaticindexcritical graphs of order 14. Our result extends that of [5] and leaves order 16 as the only case to be checked in order to decide on the minimality of the counterexample given by Chetwynd and Fiol. In addition we list all nontrivial critical graphs of order 13. Key words: critical graph, edgecolouring, graph generation. Math. Subj. Class (2001): 05C15, 05C30