Uniformly optimal graphs in some classes of graphs with node failures

The uniformly optimal graph problem with node failures consists of finding the most reliable graph in the class @W(n,m) of all graphs with n nodes and m edges in which nodes fail independently and edges never fail. The graph G is called uniformly optimal in @W(n,m) if, for all node-failure probabilities [email protected]?(0,1), the graph G is the most reliable graph in the class of graphs @W(n,m). This paper proves that the multipartite graphs K(b,b+1,...,b+1,b+2) are uniformly optimal in their classes @W((k+2)(b+1),(k^2+3k+2)(b+1)^2/2-1), where k is the number of partite sets of size (b+1), while for i>2, the multipartite graphs K(b,b+1,...,b+1,b+i) are not uniformly optimal in their classes @W((k+2)b+k+i,(k+2)(k+1)b^2/2+(k+1)(k+i)b+k(k+2i-1)/2).