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

TL;DR

This work analyzes the budget-constrained random graph process, where a player Builder observes $t$ random edges on $n$ vertices and can purchase at most $b$ of them to achieve a monotone property such as an $F$-factor or a power of a Hamilton cycle. The authors establish general lower bounds on the required budget $b$ as a function of $t$ for partial and full spanning structures and show that near the hitting time for these properties, the budget cannot be substantially smaller than the straightforward (buy-all-edges) approach; they also construct near-optimal, simple strategies achieving matching upper bounds. A key contribution is the development of multi-stage analysis tools, including a deterministic reduction for strategies and sophisticated couplings that connect stages, as well as absorption and linking techniques for powers of Hamilton cycles. Together, these results yield negative answers to longstanding questions about the budget needed near hitting times for certain spanning structures and illuminate the complexity of online graph construction under budget constraints. The methods presented—absorbers, linkages, and stage-wise couplings—have potential applicability to a broader class of multi-stage probabilistic constructions in random graphs and related combinatorial settings, advancing both theory and potential practical online 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.

Figures (3)

• Figure 1: A depiction of the optimal budget $b$ for successful $(t,b)$-strategies for $K_r$-factors when $r\in\{3,4,5,6,7\}$, as follows from \ref{['thm:factors_lower', 'thm:factor_upper']}.
• Figure 2: A depiction of the optimal budget $b$ for successful $(t,b)$-strategies for the $k$-th power of a Hamilton cycle, $k\in\{2,3,4,5\}$, as given by \ref{['thm:hamsquare_lower', 'thm:hamsquare_upper']}.
• Figure 3: The graphs $P_3^2$, $P_4^2$ and $P_5^2$.

Theorems & Definitions (60)

• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Lemma 2.1: Chernoff's bound
• Lemma 2.2: Janson's inequality
• Lemma 2.3: FKG inequality
• Lemma 3.1
• proof