On the monotonicity of the broadcast function

Authors:
Hovhannes A. Harutyunyan;Arthur L. Liestman
Affiliations:
Department of Computer Science, Concordia University, Montreal, QC, Canada, H3G 1M8;School of Computing Science, Simon Fraser University, Burnaby, BC, Canada, V5A 1S6
Venue:
Discrete Mathematics
Year:
2003

Citing 5
Cited 1

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Abstract

The broadcast function, B(n), is the minimum number of edges in any graph on n vertices such that each vertex can broadcast in time ⌉logn⌈. It is a long-standing conjecture that B(n) is non-decreasing for n in the range 2m-1 + 1 ≤ n ≤ 2m. We show that the conjecture holds for n in the range 2m-1 + 1 ≤ n ≤ 2m-1 + 2m-3. Our investigation produces a similar result for k-broadcasting, a variant of broadcasting in which an informed vertex can call up to k of its neighbors in each time unit. Along the way, we give a method to construct k-broadcast graphs, those graphs which allow minimum time k-broadcasting from each vertex for certain values of n and give some new results on t-relaxed k-broadcast graphs--those graphs which allow k-broadcasting from each vertex in minimum +t time units.