Hamiltonicity thresholds in Achlioptas processes

TL;DR

This paper investigates the thresholds for the emergence of Hamilton cycles in a selective edge-adding process, revealing three distinct regimes based on the number of candidate edges per round, and providing precise bounds for each.

Contribution

It introduces a new analysis of Hamiltonicity thresholds in a multi-edge selection process, identifying three regimes and deriving bounds for the critical points.

Findings

For K=o(log n), Hamilton cycle appears around (1+o(1))n log n / (2K).

For K=ω(log n), Hamilton cycle is achieved in n+o(n) rounds with high probability.

In the intermediate regime K=Θ(log n), the threshold is of order n with bounds differing by a factor of 3.

Abstract

In this paper we analyze the appearance of a Hamilton cycle in the following random process. The process starts with an empty graph on n labeled vertices. At each round we are presented with K=K(n) edges, chosen uniformly at random from the missing ones, and are asked to add one of them to the current graph. The goal is to create a Hamilton cycle as soon as possible. We show that this problem has three regimes, depending on the value of K. For K=o(\log n), the threshold for Hamiltonicity is (1+o(1))n\log n /(2K), i.e., typically we can construct a Hamilton cycle K times faster that in the usual random graph process. When K=\omega(\log n) we can essentially waste almost no edges, and create a Hamilton cycle in n+o(n) rounds with high probability. Finally, in the intermediate regime where K=\Theta(\log n), the threshold has order n and we obtain upper and lower bounds that differ by a multiplicative factor of 3.