Broadcasting in large unicyclic communication networks

Broadcasting is the process of information dissemination in communication networks (modelled as graphs) whereby a message originating at one vertex becomes known to all members given that at each unit of time a vertex can pass the message to at most one of its neighbours. A broadcast graph on n vertices is a network in which messages can be broadcast in the minimum possible (=[log2n]) number of steps regardless of the originator. In "Broadcasting in General Networks II: Unicyclic Graphs", (Shastri, 1998) we continued our work initiated in, "Broadcasting in General Networks I: Trees", (Shastri, 1995) by studying the problem of broadcasting in unicyclic graphs (connected graphs on n vertices having exactly n edges) as a second step towards studying broadcasting in general graphs, as opposed to the much studied problem of constructing broadcast graphs having the smallest number of edges. Unicyclic graphs of small order with the best possible broadcast time were constructed. In this paper, some of these constructions are generalised to obtain asymptotic bounds for large n and to obtain unicyclic graphs with best broadcast time of higher orders. The results make comparison with those obtained for trees in the above paper.