17000130
Removable Edges in Longest Cycles of 4-Connected Graphs
Abstract.Let G be a 4-connected graph. For an edge e of G, we do the following operations on G: first, delete the edge e from G, resulting the graph G−e; second, for all vertices x of degree 3 in G−e, delete x from G−e and then completely connect the 3 neighbors of x by a triangle. If multiple edges occur, we use single edges to replace them. The final resultant graph is denoted by G⊖e. If G⊖e is 4-connected, then e is called a removable edge of G. In this paper we obtain some results on removable edges in a longest cycle of a 4-connected graph G. We also show that for a 4-connected graph G of minimum degree at least 5 or girth at least 4, any edge of G is removable or contractible.
1990124
Removable edges in a cycle of a 4-connected graph
Let G be a 4-connected graph. For an edge e of G, we do the following operations on G: first, delete the edge e from G, resulting in the graph G − e; second, for all the vertices x of degree 3 in G − e, delete x from G − e and then completely connect the 3 neighbors of x by a triangle. If multiple edges occur, we use single edges to replace them. The final resultant graph is denoted by Ge .I fGe is still 4-connected, then e is called a removable edgeof G. In this paper, we investigate the problem on how many removable edges there are in a cycle of a 4-connected graph, and give examples to show that our results are in some sense the best possible. © 2004 Elsevier B.V. All rights reserved. MSC: 05C40; 05C38; 05C75
2785443
An upper bound on the number of edges of edge-coloring critical graphs with high maximum degree
In this paper, we prove that any edge-coloring critical graph G with maximum degree @D>=([email protected])/2, where @e= =8 and |V(G)|>[email protected]@d-4-(@D+6)/(@d-6), where @d is the minimum degree of G. It generalizes a result of Sanders and Zhao.
2242460
Deadlock-Free Multicasting in Irregular Networks Using Prefix Routing
A deadlock-free multicast scheme called prefix multicasting in irregular networks (i.e., networks with irregular topology) is studied. In prefix routing, a compact routing table is associated with each node (processor). Basically, each outgoing channel of a node is assigned a special label and an outgoing channel is selected if its label is a prefix of the label of the destination node. Node and channel labelling in an irregular network is based on a pre-defined spanning tree which may or may not be minimum. The routing process follows a two-phase process of going up and then down along the spanning tree, with a possible cross channel between two branches of the tree between two phases. It is shown that the proposed routing scheme is deadlock- and livelock-free. The approach is extended to multicasting in which the multicast packet is first forwarded up the tree to the longest common prefix (LCP) of destinations in the multicast. The packet is then treated as a multi-head worm that can split at branches of the spanning tree as the packet is sent down the tree.
2680554
Fault-tolerant adaptive and minimal routing in mesh-connected multicomputers using extended safety levels
The minimal routing problem in mesh-connected multicomputers with faulty blocks is studied. Two dimensional (2D) meshes are used to illustrate the approach. A sufficient condition for minimal routing in 2D meshes with faulty blocks is proposed. Unlike many other models that assume all the nodes know global fault distribution, our approach is based on the concept of an extended safety level which is a special form of limited fault information. Fault information is distributed to a limited number of nodes while it is still sufficient to support minimal routing. We study the existence of minimal paths at a given source node, limited distribution of fault information, and minimal routing itself. The proposed approach is also adaptive which allows all messages to use any minimal path. Our approach is the first attempt to address adaptive and minimal routing in 2D meshes with faulty blocks using limited fault information.
14750805
Deadlock-Free Routing In Irregular Networks Using Prefix Routing
We propose a deadlock-free routing scheme in irregular networks using prefix routing. Prefix routing is a special type of routing with a compact routing table associated with each node (processor). Basically, each outgoing channel of a node is assigned a special label and an outgoing channel is selected if its label is a prefix of the label of the destination node. Node and channel labeling in an irregular network is done through constructing a spanning tree. The routing process follows a two-phase process of going up and then down along the spanning tree, with a possible cross channel (shortcut) between two branches of the tree between two phases. We show that the proposed routing scheme is deadlock- and livelock-free. We also compare prefix routing with the existing up*/down* routing which has been widely used in irregular networks. Possible extensions are also discussed.
32836256
Minimum average distance of strong orientations of graphs
The average distance of a graph (strong digraph) G, denoted by µ(G) is the average, among the distances between all pairs (ordered pairs) of vertices of G. If G is a 2-edge-connected graph, then µ→min(G) is the minimum average distance taken over all strong orientations of G. A lower bound for µ→min(G) in terms of the order, size, girth and average distance of G is established and shown to be sharp for several complete multipartite graphs. It is shown that there is no upper bound for µ→min(G) in terms of µ(G). However, if every edge of G lies on 3-cycle, then it is shown that µ→min(G) ≤ 7/4 µ(G). This bound is improved for maximal planar graphs to 5/3 µ(G) and even further to 3/2 µ(G) for eulerian maximal planar graphs and for outerplanar graphs with the property that every edge lies on 3-cycle. In the last case the bound is shown to be sharp.