COUNTEREXAMPLES TO CONJECTURES FOR UNIFORMLY OPTIMALLY RELIABLE GRAPHS

  • Authors:

  • Yontha Ath;Milton Sobel

  • Affiliations:

  • Department of Mathematics, Claremont Graduate University, Claremont, California 91711-6160, yath1@csulb.edu;Department of Statistics and Applied Probability, University of California, Santa Barbara, Santa Barbara, California 94106, sobel@pstat.ucsb.edu

  • Venue:
  • Probability in the Engineering and Informational Sciences
  • Year:
  • 2000

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Abstract

In [7], several conjectures are listed about uniformly most reliable graphs, and, to date, no counterexamples have been found. These include the conjectures that an optimal reliable graph has degrees that are almost regular, has maximum girth, and has minimum diameter. In this article, we consider simple graphs and present one counterexample and another possible counterexample of these conjectures: maximum girth (i.e., maximize the length of the shortest circuit of the graph G) and minimum diameter (i.e., minimize the maximum possible distance between any pair of vertices).