SOME CONJECTURED UNIFORMLY OPTIMAL RELIABLE NETWORKS

Abstract

We consider all-terminal reliability, one of the more popular models in the field of network reliability. A graph with n nodes and e edges, where the nodes are perfectly reliable and the edges survive independently with equal probability p, is said to be a uniformly optimally reliable graph if it has for all values of p (0 ≤ p ≤ 1) an equal or higher reliability among all graphs with the same number of nodes and edges. Boesch et al. [4] verified the existence of uniformly optimally reliable graphs for e = n − 1, e = n, e = n + 1, and e = n + 2; he has also given a conjecture for e = n + 3. Wang [8] proved this conjecture. In this article, we present four new infinite families of graphs that we conjecture to be uniformly optimal.