Long paths in heterogeneous random subgraphs of graphs with large minimum degree

Abstract

For a graph Gk on n vertices with a minimum degree of at least k→∞ as n→∞, let G(n,p) be a random subgraph of Gk taken by retaining each edge (i,j) independently with probability pij and p={pij}(i,j)∈Gk. We show that under certain conditions on the edge probabilities, the resulting random graph has a long path that covers almost all or all vertices with probability tending to 1 as n→∞.