Building graphs with high minimum degree on a budget

TL;DR

This paper disproves the conjecture that any strategy using $\epsilon n$ fewer edges fails with probability bounded away from 0, and exhibits such a strategy that succeeds with probability bounded away from 0.

Abstract

We consider the problem of constructing a graph of minimum degree $k\ge 1$ in the following controlled random graph process, introduced recently by Frieze, Krivelevich and Michaeli. Suppose the edges of the complete graph on $n$ vertices are permuted uniformly at random. A player, Builder, sees the edges one by one, and must decide irrevocably upon seeing each edge whether to purchase it or not. Suppose Builder purchases an edge if and only if at least one endpoint has degree less than $k$ in her graph. Frieze, Krivelevich and Michaeli observed that this strategy succeeds in building a graph of minimum degree at least $k$ by $τ_k$, the hitting time for having minimum degree $k$. They conjectured that any strategy using $εn$ fewer edges, where $ε>0$ is any constant, fails with high probability. In this paper we disprove their conjecture. We show that for $k\ge 2$ Builder has a strategy which purchases $n/9$ fewer edges and succeeds with high probability in building a graph of minimum degree at least $k$ by $τ_k$. For $k=1$ we show that any strategy using $εn$ fewer edges fails with probability bounded away from 0, and exhibit such a strategy that succeeds with probability bounded away from 0.

Theorems & Definitions (32)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1
• Theorem 2.2: Chernoff's bound, see jlr
• Theorem 2.3: McDiarmid's inequality
• Proposition 2.4
• Theorem 2.5: Erdős and Renyi erdHos1961strength
• Theorem 2.6: Bollobás and Thomason bollobas-thomason
• Lemma 2.7