6186942
The number of removable edges in 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 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 G ⊖ e. If G ⊖ e is still 4-connected, then e is called a removable edge of G. In this paper we prove that every 4-connected graph of order at least six (excluding the 2-cyclic graph of order six) has at least (4|G|+16)/7 removable edges. We also give the structural characterization of 4-connected graphs for which the lower bound is sharp.
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.
8935539
An extended dynamic source routing scheme in ad hoc wireless networks
We consider a multipath extension to the dynamic source routing (DSR) protocol proposed by D.B. Johnson and D.A. Maltz (1996), an on-demand routing protocol, for ad hoc wireless networks. This extension keeps two node disjoint paths between the source and the destination of a routing process without introducing extra overhead. Several optimization options are also considered. Simulation is conducted on the success rate of finding node disjoint paths.
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.
12687044
A novel QoS multicast model in mobile ad hoc networks
Multicast applications for large-scale mobile ad hoc networks (MANETs) require an efficient and effective quality of service (QoS)-aware multicast model. The new requirements to guarantee QoS are high availability and good load balancing due to limited bandwidth and transmission power of mobile nodes (MNs). We propose a logical hypercube-based virtual dynamic backbone (HVDB) model for QoS-aware multicast communications. In this model, high fault tolerance and small diameter of hypercubes are the basis for high availability, and regularity and symmetry of hypercubes contribute to good load balancing. Based on this model, we design proactive local logical route maintenance, summary-based membership update, and logical location-based multicast routing algorithms in large-scale MANETs.
10342689
A fault-tolerant adaptive and minimal routing approach in n-D meshes
In this paper a sufficient condition is given for minimal routing in n-dimensional (n-D) meshes with faulty nodes contained in a set of disjoint fault regions. It is based on an early work of the author on minimal routing in low dimension meshes (such as 2-D meshes with faulty blocks). Unlike many traditional models that assume all the nodes know global fault distribution, our approach is based on the concept of limited global fault information. First, a fault model called fault region is used in which all faulty nodes in the system are contained in a set of disjoint regions. Fault information is coded in a 2n-tuple called extended safety level associated with each node of an n-D mesh to support minimal routing. Specifically, we study the existence of minimal paths at a given source node, limited distribution of fault information, minimal routing, and deadlock-free routing. Our results show that any minimal routing that is partially adaptive can still be applied as long as the destination node meets a certain safety condition. A dynamic planar-adaptive routing scheme is presented that offers better fault tolerance and adaptivity than the regular planar-adaptive routing scheme in n-D meshes.