Extremal paths in inhomogenous random graphs

Abstract

In this paper, we study long open paths in an inhomogenous Erdős–Rényi random graph G obtained from the complete graph Kn on n vertices by allowing each edge e to be open with probability pn(e), independently of other edges. If the edge probability assignment sequence satisfies certain neighbour density conditions, then G has a long path containing nearly all the vertices with high probability, i.e., with probability converging to one as n→∞. Our methods extend to random weighted graphs with nonidentical weight distributions and we describe conditions under which the minimum weight Hamiltonian path has weight bounded above by a constant, with high probability.