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 -> infinity as n -> infinity, 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)is an element of 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 -> infinity.(c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).