297940
Connectivity of Cartesian product graphs
Use v"i,@k"i,@l"i,@d"i to denote order, connectivity, edge-connectivity and minimum degree of a graph G"i for i=1,2, respectively. For the connectivity and the edge-connectivity of the Cartesian product graph, up to now, the best results are @k(G"1xG"2)>=@k"1+@k"2 and @l(G"1xG"2)>=@l"1+@l"2. This paper improves these results by proving that @k(G"1xG"2)>=min{@k"1+@d"2,@k"2+@d"1} and @l(G"1xG"2)=min{@d"1+@d"2,@l"1v"2,@l"2v"1} if G"1 and G"2 are connected undirected graphs; @k(G"1xG"2)>=min{@k"1+@d"2,@k"2+@d"1,2@k"1+@k"2,2@k"2+@k"1} if G"1 and G"2 are strongly connected digraphs. These results are also generalized to the Cartesian products of n(>=3) connected graphs and n strongly connected digraphs, respectively.
872168
The super connectivity of shuffle-cubes
The shuffle-cube SQn, where n ≡ 2 (mod 4), a new variation of hypercubes proposed by Li et al. [T.-K. Li, J.J.M. Tan, L.-H. Hsu, T.-Y. Sung, The shuffle-cubes and their generalization, Inform. Process. Lett. 77 (2001) 35-41], is an n-regular n-connected graph. This paper determines that the super connectivity of SQn is 2n - 4 and the super edge-connectivity is 2n - 2 for n ≥ 6.
7849638
Edge-fault-tolerant edge-bipancyclicity of hypercubes
In this paper, we consider the problem embedding a cycle into the hypercube Q"n with existence of faulty edges and show that for any edge subset F of Q"n with |F|==4 and all edges in F are not incident with the same vertex. This result improves some known results.
10735574
Forwarding indices of folded n-cubes
For a given connected graph G of order n, a routing R is a set of n(n - 1) elementary paths specified for every ordered pair of vertices in G. The vertex (resp. edge) forwarding index of a graph is the maximum number of paths of R passing through any vertex (resp. edge) in the graph. In this paper, the authors determine the vertex and the edge forwarding indices of a folded n-cube as (n-1)2n-1 +1-((n+l)/2) (n ⌊n+1/2⌋) and 2n-(n,⌊n+1/2⌋),respectively.
15426263
Super connectivity of line graphs
The super connectivity κ' and the super edge-connectivity λ' are more refined network reliability indices than connectivity κ and edge-connectivity λ. This paper shows that for a connected graph G with order at least four rather than a star and its line graph L(G), κ'(L(G)) = λ'(G) if and only if G is not super-λ'. As a consequence, we obtain the result of Hellwig et al. [Note on the connectivity of line graphs, Inform. Process. Lett. 91 (2004) 7] that κ(L(G)) = λ'(G). Furthermore, the authors show that the line graph of a super-λ' graph is super-λ if the minimum degree is at least three.
26138426
2-diameter of de Bruijn graphs
Binary strings are used as vertex labels or components of vertex labels. We will abbreviate “the vertex with label X” as “vertex x.” The binarji direcred de Briiijn graph of dimension n , denoted B ( n ) , has 2 ” vertices which are labeled with the binary strings of length n . There is an arc from any vertex x, x2 xn to the vertices x2x3 x,O and x2x3 x,, 1. We say that the ith coordinate of ,Y is x, . Any .u, is equal toOor l , andZ, = 1 x i . The undirected binary de Bruijn graph UB( n ) is obtained from B( n) by deleting the orientation of the arcs and omitting multiple edges and loops. It is well known that UB( n ) is 2-connected and that its diameter (maximum of the distances between all pairs of vertices) is equal to n . Due to their bounded maximum degree equal to 4 and their low diameter, de Bruijn graphs have been proposed as a possible good interconnection network for a parallel architecture [ 11. They have also been shown to solve a wide class of problems [ 21. For a k-connected graph, Hsu [ 4 ] introduced the notion of the kdiameter as follows: Given any two vertices x and y , a k-container between x and y is a set of k vertex disjoint paths between xand y . The length ofa k-container is the length of the longest of the k paths. The kdistance d k ( x , y ) between x and y is the minimum of the lengths of all the k-containers. Then, the k-diameter of a graph G is the maximum of d k ( x , y) for all pairs of vertices x and y . General results on the k-diameters of k-regular kconnected graphs can be found in [ 41 or [ 51. Results for some particular classes ofgraphs and many open problems can also be found in [ 41. It was shown by Pradhan and Reddy [ 7 ] that the 2diameter of U B ( n ) is less than or equal to 2n. The 2diameter is also obviously at least n. The aim of this paper was to prove that it is exactly n. In other words, between any two vertices of UB( n ) , there exist two internally vertex disjoint paths of length at most n. This gives a good measure of the fault tolerance of the graph [ 61. Let us first introduce some notation and recall some properties of the de Bruijn digraph B( n ) : Given two vertices x and y , we will denote f i x , y ] as a shortest path P from x to y . The length of this path, denoted by I PIX, y ] 1, is the number of edges in the path and is also the distance d ( x , y ) from x to y . P [ x , y ] also represents the set of vertices of the path, including its extremities.
7214751
A conforming finite element method for overlapping and nonmatching grids
In this paper we propose a finite element method for nonmatching overlapping grids based on the partition of unity. Both overlapping and nonoverlapping cases are considered. We prove that the new method admits an optimal convergence rate. The error bounds are in terms of local mesh sizes and they depend on neither the overlapping size of the subdomains nor the ratio of the mesh sizes from different subdomains. Our results are valid for multiple subdomains and any spatial dimensions.
5147011
Wide Diameters of Cartesian Product Graphs and Digraphs
In graph theory and study of fault tolerance and transmission delay of networks, connectivity and diameter of a graph are two very important parameters and have been deeply studied by many authors. Wide diameter combining connectivity with diameter is a more important parameter to measure fault tolerance and efficiency of parallel processing computer networks and has received much attention in the recent years. Diameter with width k of a graph G is defined as the minimum integer d for which between any two distinct vertices in G there exist at least k internally disjoint paths of length at most d. In the present paper, the tight upper bounds of wide diameter of the Cartesian product graphs are obtained. Some known results can be deduced or improved from ours.
11486473
Circumference of Graphs with Bounded Degree
Karger, Motwani, and Ramkumar Algorithmica, 18 (1997), pp. 82--98] have shown that there is no constant approximation algorithm to find a longest cycle in a Hamiltonian graph, and they conjectured that this is the case even for graphs with bounded degree. On the other hand, Feder, Motwani, and Subi [SIAM J. Comput., 31 (2002), pp. 1596--1607] have shown that there is a polynomial time algorithm for finding a cycle of length $n^{\log_32}$ in a 3-connected cubic n-vertex graph. In this paper, we show that if G is a 3-connected n-vertex graph with maximum degree at most d, then one can find, in O(n3) time, a cycle in G of length at least $\Omega(n^{\log_b2})$, where $b=2(d-1)^2+1$.
14297439
Geometric permutations of high dimensional spheres
We prove the maximum number of geometric permutations, induced by line transversals to a set of <i>n</i> pairwise disjoint congruent spheres in <i>R<sup>d</sup></i> with <i>d</i> ⪈ 3, is no more than 4 when <i>n</i> is sufficiently large, achieving the best known upper bound for this problem. We also prove the maximum number of geometric permutations of a set of <i>n</i> noncongruent spheres of bounded radius ratio in R<sup>d</sup>, <i>d</i> ⪈ 3, is at most 2<sup>[√2<i>M</i>]+1</sup>, where <i>M</i> is the ratio or the largest radius and the smallest radius. Our result settles a conjecture in combinatorial geometry.
7149198
A fault-tolerant approach to secure information retrieval
Several private information retrieval (PIR) schemes were proposed to protect users' privacy when sensitive information stored in database servers is retrieved. However, existing PIR schemes assume that any attack to the servers does not change the information stored and any computational results. We present a novel fault-tolerant PIR scheme (called FT-PIR) that protects users' privacy and at the same time ensures service availability in the presence of malicious server faults. Our scheme neither relies on any unproven cryptographic assumptions nor the availability of tamper-proof hardware. A probabilistic verification function is introduced into the scheme to detect corrupted results. Unlike previous PIR research that attempted mainly to demonstrate the theoretical feasibility of PIR, we have actually implemented both a PIR scheme and our FT-PIR scheme in a distributed database environment. The experimental and analytical results show that only modest performance overhead is introduced by FT-PIR while comparing with PIR in the fault-free cases. The FT-PIR scheme tolerates a variety of server faults effectively. In certain fail-stop fault scenarios, FT-PIR performs even better than PIR. It was observed that 35.82% less processing time was actually needed for FT-PIR to tolerate one server fault.