On constructing small subgraphs in the budget-constrained random graph process

TL;DR

This paper determines the minimal budget needed to reliably construct small fixed subgraphs like K4 and wheels in a random graph process, providing optimal strategies and bounds.

Contribution

It introduces optimal strategies and tight bounds for constructing specific small subgraphs under budget constraints in a random graph process.

Findings

Optimal budget for constructing K4 is ( max ight ext{,}n^8/t^5,n^2/t)

Achieves tight bounds for wheels and certain trees plus universal vertices

Provides bounds for K5 with some remaining gaps

Abstract

Consider the budget-constrained random graph process introduced by Frieze, Krivelevich and Michaeli, where each time an edge is offered through the (standard) random graph process we must irrevocably decide whether to "purchase" this edge or not, with our goal being to construct a graph which satisfies some property within a given time $t$ and while purchasing at most $b$ edges. We consider the problem of constructing graphs containing certain fixed small subgraphs. We provide an optimal strategy for building a graph which contains a copy of $K_4$, showing that budget $b=\omega(\max\{n^8/t^5,n^2/t\})$ suffices and that if $b=o(\max\{n^8/t^5,n^2/t\})$ then no strategy can a.a.s. produce a graph containing a copy of $K_4$. This resolves a problem raised by I\v{l}kovi\v{c}, Le\'{o}n and Shu. More generally, we obtain analogously tight results for containing a wheel of any fixed size, or a graph consisting of a tree plus one additional universal vertex. We also tackle the problem of constructing graphs containing a copy of $K_5$, obtaining both lower and upper bounds on the optimal budget, though a gap remains in this case.