d-connectivity of the random graph with restricted budget

In this short note, we consider a graph process recently introduced by Frieze, Krivelevich and Michaeli. In their model, the edges of the complete graph K-n are ordered uniformly at random and are then revealed consecutively to a player called Builder. At every round, Builder must decide if they accept the edge proposed at this round or not. We prove that, for every d >= 2, Builder can construct a spanning d-connected graph after (1 + o (1)) n log n/2 rounds by accepting (1 + o (1)) dn/2 edges with probability converging to 1 as n -> 8. This settles a conjecture of Frieze, Krivelevich and Michaeli.