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

TL;DR

The paper tackles the Hamiltonicity problem in the semi-random graph process by developing a fully adaptive three-stage strategy that achieves a Hamiltonian cycle in $αn$ rounds with $α<1.81696$, while proving a nontrivial lower bound $βn$ with $β>1.26575$. The authors employ the differential equation method to analyze a suite of evolving random variables under an intricate coloring and augmentation scheme, capturing the process through a coupled system of ODEs. Their upper-bound construction combines a degree-greedy preparatory phase, a fully adaptive randomized phase guided by the ODEs, and a precise cleanup step that absorbs the remaining vertices in sublinear time, yielding a rigorous $O(n)$ bound with explicit constants. On the flip side, they derive a combinatorial lower bound by counting potential Hamiltonian-contributing structures and applying concentration arguments to show that certain configurations prevent Hamiltonicity before a linear number of rounds, establishing $β$ as a root of a carefully defined function. Collectively, the results tighten the threshold gap for Hamiltonicity in the semi-random process and illustrate the power of adaptive strategies guided by differential equations in combinatorial processes.

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 $αn$ rounds, where $α< 1.81696$ is derived from the solution to a system of differential equations. We also show that achieving Hamiltonicity requires at least $βn$ rounds, where $β> 1.26575$.

Theorems & Definitions (29)

• Remark 1
• Theorem 1.1
• Theorem 1.2
• Lemma 2.1: Boundedness Hypothesis -- \ref{['alg:fully_randomized']}
• proof
• Lemma 2.2: Trend Hypothesis -- \ref{['alg:fully_randomized']}
• proof
• Lemma 2.3: Concentration of \ref{['alg:fully_randomized']}'s Random Variables
• Proposition 2.4
• Lemma 2.5: Clean-up Algorithm