Graph factors and powers of Hamilton cycles in the budget-constrained random graph process

TL;DR

This paper investigates the minimum budget needed for a player in a random graph process to construct certain spanning structures, providing bounds, strategies, and new analytical tools for properties like $F$-factors and powers of Hamilton cycles.

Contribution

It establishes tight lower bounds on the budget for constructing spanning structures in a budget-constrained random graph process and introduces novel tools for analyzing multi-stage strategies.

Findings

Lower bounds on budget $b$ as a function of $t$ for partial $F$-factors

A simple strategy that is nearly optimal for constructing $F$-factors

Development of new tools for analyzing multi-stage strategies

Abstract

We consider the following budget-constrained random graph process introduced by Frieze, Krivelevich and Michaeli. A player, called Builder, is presented with $t$ distinct edges of $K_n$ one by one, chosen uniformly at random. Builder may purchase at most $b$ of these edges, and must (irrevocably) decide whether to purchase each edge as soon as it is offered. Builder's goal is to construct a graph which satisfies a certain property; we investigate the properties of containing different $F$-factors or powers of Hamilton cycles. We obtain general lower bounds on the budget $b$, as a function of $t$, required for Builder to obtain partial $F$-factors, for arbitrary $F$. These imply lower bounds for many distinct spanning structures, such as powers of Hamilton cycles. Notably, our results show that, if $t$ is close to the hitting time for a partial $F$-factor, then the budget $b$ cannot be substantially lower than $t$. These results give negative answers to questions of Frieze, Krivelevich and Michaeli. Conversely, we also exhibit a simple strategy for constructing (partial) $F$-factors, in particular showing that our general lower bound is tight up to constant factors. The ideas from this strategy can be exploited for other properties. As an example, we obtain an essentially optimal strategy for powers of Hamilton cycles. In order to formally prove that this strategy succeeds, we develop novel tools for analysing multi-stage strategies, which may be of general interest for studying other properties.