Threshold dominating cliques in random graphs and interval routing

The existence of (shortest-path) interval routing schemes for random graphs that use at most one interval label per edge is an open problem posed in [C. Gavoille, D. Peleg, The compactness of interval routing for almost all graphs, SIAM J. Computing 31 (3) (2001) 706-721]. In this paper, we show that for any random graph G(n,p) with edge probability p>0.765, there exists an interval routing scheme that uses at most one label per edge and has an additive stretch 1. In doing so, we provide an interesting construction of such an interval routing scheme for graphs that have a 12-threshold dominating clique, and establish a general result on the existence of threshold dominating cliques in random graphs.