Building graphs with high minimum degree on a budget

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 add it to her graph (of already selected edges) or not. Suppose Builder decides to add an edge to her graph 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 ${\tau }_k$, the hitting time for having minimum degree $k$. They conjectured that any strategy using $\varepsilon n$ fewer edges, where $\varepsilon >0$ 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 ${\tau }_k$. For $k=1$ we show that any strategy using $\varepsilon n$ fewer edges fails with probability bounded away from 0, and exhibit such a strategy that succeeds with probability bounded away from 0.