$d$-connectivity of the random graph with restricted budget

Abstract

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 $d\ge 2$, 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 $n\to \infty$. This settles a conjecture of Frieze, Krivelevich and Michaeli.