Building Hamiltonian Cycles in the Semi-Random Graph Process in Less Than $2n$ Rounds

TL;DR

This paper introduces an adaptive strategy for the semi-random graph process that constructs Hamiltonian cycles in fewer than 2n rounds, improving understanding of the process's efficiency in achieving Hamiltonicity.

Contribution

The paper presents a new adaptive strategy that creates Hamiltonian cycles in less than 1.81696n rounds and establishes a lower bound of over 1.26575n rounds for the process.

Findings

Hamiltonian cycle constructed in less than 1.81696n rounds

Lower bound of more than 1.26575n rounds for Hamiltonicity

Differential equations used to derive bounds

Abstract

The semi-random graph process is an adaptive random graph process in which an online algorithm is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the algorithm independently and uniformly at random. The algorithm then adaptively selects a vertex $v$, and adds the edge $uv$ to the graph. For a given graph property, the objective of the algorithm is to force the graph to satisfy this property asymptotically almost surely in as few rounds as possible. We focus on the property of Hamiltonicity. We present an adaptive strategy which creates a Hamiltonian cycle in $\alpha n$ rounds, where $\alpha < 1.81696$ is derived from the solution to a system of differential equations. We also show that achieving Hamiltonicity requires at least $\beta n$ rounds, where $\beta > 1.26575$.