110 research outputs found
Decomposing the cube into paths
We consider the question of when the -dimensional hypercube can be
decomposed into paths of length . Mollard and Ramras \cite{MR2013} noted
that for odd it is necessary that divides and that . Later, Anick and Ramras \cite{AR2013} showed that these two conditions are
also sufficient for odd and conjectured that this was true for
all odd . In this note we prove the conjecture.Comment: 7 pages, 2 figure
On the spanning tree packing number of a graph: a survey
AbstractThe spanning tree packing number or STP number of a graph G is the maximum number of edge-disjoint spanning trees contained in G. We use an observation of Paul Catlin to investigate the STP numbers of several families of graphs including quasi-random graphs, regular graphs, complete bipartite graphs, cartesian products and the hypercubes
Properties and algorithms of the hyper-star graph and its related graphs
The hyper-star interconnection network was proposed in 2002 to overcome the
drawbacks of the hypercube and its variations concerning the network cost, which is
defined by the product of the degree and the diameter. Some properties of the graph
such as connectivity, symmetry properties, embedding properties have been studied
by other researchers, routing and broadcasting algorithms have also been designed.
This thesis studies the hyper-star graph from both the topological and algorithmic
point of view. For the topological properties, we try to establish relationships between
hyper-star graphs with other known graphs. We also give a formal equation for the
surface area of the graph. Another topological property we are interested in is the
Hamiltonicity problem of this graph.
For the algorithms, we design an all-port broadcasting algorithm and a single-port
neighbourhood broadcasting algorithm for the regular form of the hyper-star graphs.
These algorithms are both optimal time-wise.
Furthermore, we prove that the folded hyper-star, a variation of the hyper-star, to be
maixmally fault-tolerant
Decompositions of -Cube into -Cycles
It is known that the -dimensional hypercube for even, has a
decomposition into -cycles for with In
this paper, we prove that has a decomposition into -cycles for As an immediate consequence of this result, we get path
decompositions of as well. This gives a partial solution to a conjecture
posed by Ramras and also, it solves some special cases of a conjecture due to
Erde
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