42869793
Edge-coloring critical graphs with high degree
It is proved here that any edge-coloring critical graph of order n and maximum degree Δ ≥ 8 has the size at least 3(n + Δ - 8). It generalizes a result of Hugh Hind and Yue Zhao.
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.
15322039
Tree coloring of distance graphs with a real interval set
Let RR be the set of real numbers and DD be a subset of the positive real numbers. The distance graph  G(R,D)G(R,D) is a graph with the vertex set RR and two vertices xx and yy are adjacent if and only if |x−y|∈D|x−y|∈D. In this work, the vertex arboricity (i.e., the minimum number of subsets into which the vertex set V(G)V(G) can be partitioned so that each subset induces an acyclic subgraph) of G(R,D)G(R,D) is determined for DD being an interval between 1 and δδ.
45978330
Two sufficient conditions for a graph to be type 1
1502406
Total Coloring of Series-Parallel Graphs
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.
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.
42778533
An Optimal Fault-Tolerant Nonredundant Broadcasting Scheme in Injured Hypercubes
Abstract We propose a nonredundant broadcasting scheme for injured hypercube multicomputers with direct-connection capability, that is, where each hypercube node has a separate router. The proposed scheme is first presented to cover faulty links only. Under the assumption that no more than n − 1 faulty links exist in an injured n-cube, the proposed scheme is optimal in the sense that it generates a spanning binomial tree from a given node whenever there exists such a tree originated from that node. The scheme is also extended to cover both node and link faults. Although the extended scheme is no longer optimal, we show that by using the extended scheme the maximum number of routing steps required from the source node s to any other node a is H(s, a) + 2, where H stands for the Hamming distance. The proposed scheme uses global network information. Evaluation studies show that the proposed scheme achieves optimal broadcasting a high percentage of times.
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.
18143775
An extended link reversal protocol in dynamic networks
We consider the problem of maintaining routing paths between nodes in a dynamic network. Gafni and Bertsekas proposed a link reversal approach called the BG method that maintains a directed acyclic graph (DAG) with a given destination as the sink node. By virtue of built-in redundancy, an updating algorithm to establish a new DAG is activated infrequently and it happens only when the last outgoing link of a host in the DAG is destroyed due to the movement of nodes. In this paper, we propose another updating approach that tries to minimize the total number of reversed links and to maintain routing information without using much extra overhead. The approach maintains a reversed breadth-first tree. Nodes in the network are either marked (inside the tree) or unmarked (outside the tree). When it is too costly to maintain a minimum path for a marked node, the branch rooted at the node is trimmed and the approach then gracefully switches to the BG method. Several extensions are also discussed. A simulation study is conducted to compare the performance of the proposed approach with the existing one.
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.