Discrete Applied Mathematics 37,‘38 (1992) 297-3 17 North-Holknd 297 Broadcasting and span ing trees in de Bruijn and K;;lutz networ J. Opatrny D. Sotteau LRI. UA 410 CNRS. hit 490, UtriversitP de Par&Sud, 91405 Orsoy Cedex, Fratwt: Received 6 November 1989 Revised 8 April 1990 Abstract Heydemann, M.C., J. Opatmy and D. Sotteau, BroaJcasting and spanning trees in de Bruijn and Kautz networks, Discrete Applied Mathematics 37138 ( 1992) 297-3 17. We prc e that. for any p I d. there exists a spanning directed p-ary tree of depth at most D [log,, zyxwvutsrqponmlkjih (11 in a de Bruijn digraph B(d, D) or in a Kautz digraph K(d, Dj of degree dand diameter D. This result gives directly an upper bound of pD rlo$ dl on the broadcast time of these digraphs, which improves the previously known bounds for d L 15. In the case of de Bruijn digraphs, an upper bound on the broad- cast time of B@q, D) in terms of the broadcast times of B(p, D) and B(q, D) is established. This is used to improve the upper bounds on the broadcast time of B(d, D). We obtain several results which are refinements of the follov4rq general statements: fi)foranyDz2,dz2,if2” < d I 2k, b(B(d, D)) 5 (2 k + 3)D. (ii) for any k L 3, if 2’ ’ * d I 2k, 2k - I 5 b(B(d, 2)) I 2k. 1. Introduction One of the common processes in communication networks is the sending of a message from one node of a network to all the other nodes as quickly as possible, subject to the following constraint: during each unit of time a n>de which already * The work was supported partially by NSERC of Canada and by PRC C3 ot’ France and was done i%cr.ile the third author was visiting McGill University. 0166-218X/92/$05.0 /c l992-Elsevier Science Publishers B.V. All rights reserved