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

TL;DR

This paper proves that in a random graph process, a player can efficiently build a spanning d-connected graph with high probability, confirming a conjecture about the process's capabilities.

Contribution

It establishes that Builder can construct a spanning d-connected graph in the random process within near-optimal rounds, settling a conjecture by Frieze, Krivelevich, and Michaeli.

Findings

Builder can create a spanning d-connected graph after approximately n log n/2 rounds.

Builder accepts approximately dn/2 edges with high probability.

The result confirms the conjecture for all d ≥ 2.

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.