Weak saturation numbers in random graphs

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 Erd\H{o}s--R\'enyi 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.