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Weak saturation numbers in random graphs

Olga Kalinichenko, Meysam Miralaei, Ali Mohammadian, Behruz Tayfeh-Rezaie

Abstract

For two given graphs $G$ and $F$, a graph $ H$ is said to be weakly $ (G, F) $-saturated if $H$ is a spanning subgraph of $ G$ which has no copy of $F$ as a subgraph and one can add all edges in $ E(G)\setminus E(H)$ to $ H$ in some order so that a new copy of $F$ is created at each step. The weak saturation number $ wsat(G, F)$ is the minimum number of edges of a weakly $(G, F)$-saturated graph. In this paper, we deal with the relation between $ wsat(G(n,p), F)$ and $ wsat(K_n, F)$, where $G(n,p)$ denotes the Erdős--Rényi random graph and $ K_n$ denotes the complete graph on $ n$ vertices. For every graph $ F$ and constant $ p$, we prove that $ wsat( G(n,p),F)= wsat(K_n,F)(1+o(1))$ with high probability. Also, for some graphs $ F$ including complete graphs, complete bipartite graphs, and connected graphs with minimum degree $ 1$ or $ 2$, it is shown that there exists an $ \varepsilon(F)>0$ such that, for any $ p\geqslant n^{-\varepsilon(F)}\log n$, $ wsat( G(n,p),F)= wsat(K_n,F)$ with high probability.

Weak saturation numbers in random graphs

Abstract

For two given graphs and , a graph is said to be weakly -saturated if is a spanning subgraph of which has no copy of as a subgraph and one can add all edges in to in some order so that a new copy of is created at each step. The weak saturation number is the minimum number of edges of a weakly -saturated graph. In this paper, we deal with the relation between and , where denotes the Erdős--Rényi random graph and denotes the complete graph on vertices. For every graph and constant , we prove that with high probability. Also, for some graphs including complete graphs, complete bipartite graphs, and connected graphs with minimum degree or , it is shown that there exists an such that, for any , with high probability.
Paper Structure (4 sections, 23 theorems, 32 equations)

This paper contains 4 sections, 23 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Asymptotic stability
  3. Exact stability
  4. Small $\boldsymbol{p}$

Key Result

Theorem 1.1

Let $F$ be a graph with $\delta(F) \geqslant 1$. Also, let $p \in (0,1)$ and $c\geqslant\delta(F)-1$ be constants. For every positive integer $n$, assume that there exists a weakly $(K_n, F)$-saturated graph $H_n$ containing a set of vertices $S_n$ with $|S_n| \leqslant c$ such that each vertex from

Theorems & Definitions (36)

  • Theorem 1.1: Kalinichenko
  • Corollary 1.2: Kalinichenko
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1: kahnRiordan
  • Lemma 2.2
  • proof
  • Lemma 2.3: Spencer
  • Corollary 2.4
  • proof
  • ...and 26 more