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

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 pi j and p = {pi j}(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 → ∞.