549 research outputs found
Connectivity Threshold for random subgraphs of the Hamming graph
We study the connectivity of random subgraphs of the -dimensional Hamming
graph , which is the Cartesian product of complete graphs on
vertices. We sample the random subgraph with an i.i.d.\ Bernoulli bond
percolation on with parameter . We identify the window of the
transition: when the probability that the graph is
connected goes to , while when it converges to
.
We also investigate the connectivity probability inside the critical window,
namely when .
We find that the threshold does not depend on , unlike the phase
transition of the giant connected component the Hamming graph (see [Bor et al,
2005]). Within the critical window, the connectivity probability does depend on
d. We determine how.Comment: 10 pages, no 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
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