Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Sylwia Antoniuk, Alberto Espuny Díaz, Kalina Petrova, Miloš Stojaković

TL;DR

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.s.a. produce a graph containing a copy of $K_4$.

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 $b=ω(\max\{n^8/t^5,n^2/t\})$ 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 Iľkovič, León 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.

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

TL;DR

An optimal strategy for building a graph which contains a copy of , showing that budget suffices and that if then no strategy can a.s.a. produce a graph containing a copy of .

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 and while purchasing at most 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 , showing that budget suffices and that if then no strategy can a.a.s. produce a graph containing a copy of . This resolves a problem raised by Iľkovič, León 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 , obtaining both lower and upper bounds on the optimal budget, though a gap remains in this case.
Paper Structure (8 sections, 16 theorems, 81 equations, 3 figures, 6 algorithms)

This paper contains 8 sections, 16 theorems, 81 equations, 3 figures, 6 algorithms.

Table of Contents

  1. Introduction
  2. Notation and tools
  3. Lower bounds
  4. Upper bounds
  5. Concluding remarks and open problems
  6. Lower bounds
  7. Upper bounds
  8. Proof of Theorem 1.2

Key Result

Theorem 1.1

Let $k\geq4$ be an integer. For all $t\in[M]$, if then for any $(t,b)$-strategy a.a.s. $B_t$ does not contain a copy of $W_k$. On the other hand, if then there exists a successful $(t,b)$-strategy for constructing a copy of $W_k$.

Figures (3)

  • Figure 1: A depiction of the optimal budget $b$ for successful $(t,b)$-strategies for constructing copies of $W_k$ with $k\in\{4,5,6,7\}$, as given by \ref{['thm:wheels_main']}.
  • Figure 2: A depiction of the optimal budget $b$ for successful $(t,b)$-strategies for $K_4$, as given by \ref{['thm:wheels_main']}, compared with the optimal budget for its cyclic subgraphs ($K_3$, $K_3^+$ and $C_4$ are due to FKM25, and $K_4^-$ is due to ILS24). Additional depiction of the upper bound and lower bound on the optimal budget $b$ for successful $(t,b)$-strategies for $K_5$, as given by \ref{['thm:K5_main']}.
  • Figure 3: A depiction of the upper bound and lower bound on the optimal budget $b$ for successful $(t,b)$-strategies for $K_6$. The lower bound follows from \ref{['lem:max_no_copies']}, and the proof of the upper bound is omitted.

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Lemma 2.1: Chernoff bound
  • Lemma 2.2
  • proof
  • Lemma 2.3: EGNS25a
  • Lemma 3.1
  • proof
  • ...and 16 more