Disjoint Rooted Spanning Trees with Small Depths in deBruijn and Kautz Graphs

The problem of broadcasting long messages on store-and-forward communication networks, where a processor (node) can send and receive messages simultaneously to and from all its neighbors, was studied by Bermond and Fraigniaud. In such networks, the delays encountered by a message from a node $v$ to all other nodes over a broadcast spanning tree is directly proportional to the length of the paths in the tree over which the message is sent. Furthermore, the speed of the broadcast can be improved by the segmentation of the message at $v$ into equal-length segments and then the broadcast of these segments over arc-disjoint broadcast spanning trees simultaneously. These observations lead Bermond and Fraigniaud to look for the maximum number of arc-disjoint spanning trees in a deBruijn network rooted at an arbitrary node with small depths. This paper improves and extends the results of the above authors.