. In this paper, we introduce an on-line protocol which routes any set of packets along shortest paths through an arbitrary N-node 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 bounded-degree 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 ...

We introduce an on-line 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 bounded-degree 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 non-shortest paths, deterministic routing, or routing with bounded buffers.

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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

A chromatic-index-critical graph G on n vertices is non-trivial if it has at most \Deltab n 2 c edges. We prove that there is no chromatic-index-critical graph of order 12, and that there are precisely two non-trivial chromatic index critical graphs on 11 vertices. Together with known results this implies that there are precisely three non-trivial 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...

We introduce an on-line 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-degree 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 non-shortest 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...

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A k-critrical 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 k-critical graph is of odd order. Fiorini and Wilson [6] conjectured that every k-critical graph of even order has a 1-factor. Chetwynd and Yap [4] stated the problem whether it is true that if G is a k-critical graph of odd order, then G \Gamma v has a 1-factor 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 k-critical 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...