k-Connectedness and k-Factors in the Semi-Random Graph Process

Abstract

The semi-random graph process is a single-player graph game where the player is initially presented an edgeless graph with n vertices. In each round, the player is offered a vertex u uniformly at random and subsequently chooses a second vertex v deterministically ac- cording to some strategy, and adds edge uv to the graph. The objective for the player is then to ensure that the graph fulfils some specified property as fast as possible.

We investigate the properties of being k-connected and containing a k-factor. We settle the open case for 2-connectedness by showing that the player has a strategy to construct a 2-connected graph asymptotically almost surely in (ln 2+ln(ln 2+1)+o(1))n rounds, which matches a known lower bound asymptotically. We also provide a strategy for building a k-factor asymptotically almost surely in (β + 10^−5)n rounds, where β is derived from the solution of a system of differential equations.

Additionally, we consider a variant that was recently suggested by Wormald where the player chooses the first vertex and the second vertex is chosen uniformly at random. We show that the bounds for k-connectedness for the traditional setting are also tight for this variant.