18 research outputs found
Finding a Maximum 2-Matching Excluding Prescribed Cycles in Bipartite Graphs
We introduce a new framework of restricted 2-matchings close to Hamilton cycles. For an undirected graph (V,E) and a family U of vertex subsets, a 2-matching F is called U-feasible if, for each setU in U, F contains at most |setU|-1 edges in the subgraph induced by U. Our framework includes C_{= 5. For instance, in bipartite graphs in which every cycle of length six has at least two chords, our algorithm solves the maximum C_{<=6}-free 2-matching problem in O(n^2 m) time, where n and m are the numbers of vertices and edges, respectively
On prisms, M\"obius ladders and the cycle space of dense graphs
For a graph X, let f_0(X) denote its number of vertices, d(X) its minimum
degree and Z_1(X;Z/2) its cycle space in the standard graph-theoretical sense
(i.e. 1-dimensional cycle group in the sense of simplicial homology theory with
Z/2-coefficients). Call a graph Hamilton-generated if and only if the set of
all Hamilton circuits is a Z/2-generating system for Z_1(X;Z/2). The main
purpose of this paper is to prove the following: for every s > 0 there exists
n_0 such that for every graph X with f_0(X) >= n_0 vertices, (1) if d(X) >=
(1/2 + s) f_0(X) and f_0(X) is odd, then X is Hamilton-generated, (2) if d(X)
>= (1/2 + s) f_0(X) and f_0(X) is even, then the set of all Hamilton circuits
of X generates a codimension-one subspace of Z_1(X;Z/2), and the set of all
circuits of X having length either f_0(X)-1 or f_0(X) generates all of
Z_1(X;Z/2), (3) if d(X) >= (1/4 + s) f_0(X) and X is square bipartite, then X
is Hamilton-generated. All these degree-conditions are essentially
best-possible. The implications in (1) and (2) give an asymptotic affirmative
answer to a special case of an open conjecture which according to [European J.
Combin. 4 (1983), no. 3, p. 246] originates with A. Bondy.Comment: 33 pages; 5 figure
Interconnection networks for parallel and distributed computing
Parallel computers are generally either shared-memory machines or distributed- memory machines. There are currently technological limitations on shared-memory architectures and so parallel computers utilizing a large number of processors tend tube distributed-memory machines. We are concerned solely with distributed-memory multiprocessors. In such machines, the dominant factor inhibiting faster global computations is inter-processor communication. Communication is dependent upon the topology of the interconnection network, the routing mechanism, the flow control policy, and the method of switching. We are concerned with issues relating to the topology of the interconnection network. The choice of how we connect processors in a distributed-memory multiprocessor is a fundamental design decision. There are numerous, often conflicting, considerations to bear in mind. However, there does not exist an interconnection network that is optimal on all counts and trade-offs have to be made. A multitude of interconnection networks have been proposed with each of these networks having some good (topological) properties and some not so good. Existing noteworthy networks include trees, fat-trees, meshes, cube-connected cycles, butterflies, Möbius cubes, hypercubes, augmented cubes, k-ary n-cubes, twisted cubes, n-star graphs, (n, k)-star graphs, alternating group graphs, de Bruijn networks, and bubble-sort graphs, to name but a few. We will mainly focus on k-ary n-cubes and (n, k)-star graphs in this thesis. Meanwhile, we propose a new interconnection network called augmented k-ary n- cubes. The following results are given in the thesis.1. Let k ≥ 4 be even and let n ≥ 2. Consider a faulty k-ary n-cube Q(^k_n) in which the number of node faults f(_n) and the number of link faults f(_e) are such that f(_n) + f(_e) ≤ 2n - 2. We prove that given any two healthy nodes s and e of Q(^k_n), there is a path from s to e of length at least k(^n) - 2f(_n) - 1 (resp. k(^n) - 2f(_n) - 2) if the nodes s and e have different (resp. the same) parities (the parity of a node Q(^k_n) in is the sum modulo 2 of the elements in the n-tuple over 0, 1, ∙∙∙ , k - 1 representing the node). Our result is optimal in the sense that there are pairs of nodes and fault configurations for which these bounds cannot be improved, and it answers questions recently posed by Yang, Tan and Hsu, and by Fu. Furthermore, we extend known results, obtained by Kim and Park, for the case when n = 2.2. We give precise solutions to problems posed by Wang, An, Pan, Wang and Qu and by Hsieh, Lin and Huang. In particular, we show that Q(^k_n) is bi-panconnected and edge-bipancyclic, when k ≥ 3 and n ≥ 2, and we also show that when k is odd, Q(^k_n) is m-panconnected, for m = (^n(k - 1) + 2k - 6’ / ‘_2), and (k -1) pancyclic (these bounds are optimal). We introduce a path-shortening technique, called progressive shortening, and strengthen existing results, showing that when paths are formed using progressive shortening then these paths can be efficiently constructed and used to solve a problem relating to the distributed simulation of linear arrays and cycles in a parallel machine whose interconnection network is Q(^k_n) even in the presence of a faulty processor.3. We define an interconnection network AQ(^k_n) which we call the augmented k-ary n-cube by extending a k-ary n-cube in a manner analogous to the existing extension of an n-dimensional hypercube to an n-dimensional augmented cube. We prove that the augmented k-ary n-cube Q(^k_n) has a number of attractive properties (in the context of parallel computing). For example, we show that the augmented k-ary n-cube Q(^k_n) - is a Cayley graph (and so is vertex-symmetric); has connectivity 4n - 2, and is such that we can build a set of 4n - 2 mutually disjoint paths joining any two distinct vertices so that the path of maximal length has length at most max{{n- l)k- (n-2), k + 7}; has diameter [(^k) / (_3)] + [(^k - 1) /( _3)], when n = 2; and has diameter at most (^k) / (_4) (n+ 1), for n ≥ 3 and k even, and at most [(^k)/ (_4) (n + 1) + (^n) / (_4), for n ^, for n ≥ 3 and k odd.4. We present an algorithm which given a source node and a set of n - 1 target nodes in the (n, k)-star graph S(_n,k) where all nodes are distinct, builds a collection of n - 1 node-disjoint paths, one from each target node to the source. The collection of paths output from the algorithm is such that each path has length at most 6k - 7, and the algorithm has time complexity O(k(^3)n(^4))
Automorphisms generating disjoint Hamilton cycles in star graphs
In the first part of the thesis we define an automorphism φn for each star graph
Stn of degree n − 1, which yields permutations of labels for the edges of Stn
taken from the set of integers {1, . . . , bn/2c}. By decomposing these permutations
into permutation cycles, we are able to identify edge-disjoint Hamilton cycles
that are automorphic images of a known two-labelled Hamilton cycle H1 2(n)
in Stn. Our main result is an improvement from the existing lower bound of
bϕ(n)/10c to b2ϕ(n)/9c, where ϕ is Euler’s totient function, for the known number
of edge-disjoint Hamilton cycles in Stn for all odd integers n. For prime n, the
improvement is from bn/8c to bn/5c. We extend this result to the cases when n
is the power of a prime other than 3 and 7.
The second part of the thesis studies ‘symmetric’ collections of edge-disjoint
Hamilton cycles in Stn, i.e. collections that comprise images of H1 2(n) under
general label-mapping automorphisms. We show that, for all even n, there exists
a symmetric collection of bϕ(n)/2c edge-disjoint Hamilton cycles, and Stn cannot
have symmetric collections of greater than bϕ(n)/2c such cycles for any n. Thus,
Stn is not symmetrically Hamilton decomposable if n is not prime. We also give
cases of even n, in terms of Carmichael’s reduced totient function λ, for which
‘strongly’ symmetric collections of edge-disjoint Hamilton cycles, which are generated
from H1 2(n) by a single automorphism, can and cannot attain the optimum
bound bϕ(n)/2c for symmetric collections. In particular, we show that if n is a
power of 2, then Stn has a spanning subgraph with more than half of the edges
of Stn, which is strongly symmetrically Hamilton decomposable. For odd n, it remains
an open problem as to whether the bϕ(n)/2c can be achieved for symmetric
collections, but we are able to show that, for certain odd n, a ϕ(n)/4 bound is
achievable and optimal for strongly symmetric collections.
The search for edge-disjoint Hamilton cycles in star graphs is important for the
design of interconnection network topologies in computer science. All our results
improve on the known bounds for numbers of any kind of edge-disjoint Hamilton
cycles in star graphs
Finding a Maximum Restricted -Matching via Boolean Edge-CSP
The problem of finding a maximum -matching without short cycles has
received significant attention due to its relevance to the Hamilton cycle
problem. This problem is generalized to finding a maximum -matching which
excludes specified complete -partite subgraphs, where is a fixed
positive integer. The polynomial solvability of this generalized problem
remains an open question. In this paper, we present polynomial-time algorithms
for the following two cases of this problem: in the first case the forbidden
complete -partite subgraphs are edge-disjoint; and in the second case the
maximum degree of the input graph is at most . Our result for the first
case extends the previous work of Nam (1994) showing the polynomial solvability
of the problem of finding a maximum -matching without cycles of length four,
where the cycles of length four are vertex-disjoint. The second result expands
upon the works of B\'{e}rczi and V\'{e}gh (2010) and Kobayashi and Yin (2012),
which focused on graphs with maximum degree at most . Our algorithms are
obtained from exploiting the discrete structure of restricted -matchings and
employing an algorithm for the Boolean edge-CSP.Comment: 20 pages, 2 figure