Weak saturation numbers in random graphs

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

The weak saturation number of K2,t\boldsymbol{K_{2, t}}

For two graphs $G$ and $F$, we say that $G$ is weakly $F$-saturated if $G$ contains no copy of $F$ as a subgraph and one could join all the nonadjacent pairs of vertices of $G$ in some order so that a new copy of $F$ is created at each step. The weak saturation number $\mathrm{wsat}(n, F)$ is the minimum number of edges of a weakly $F$-saturated graph on $n$ vertices. In this paper, we examine $\mathrm{wsat}(n, K_{s, t})$, where $K_{s, t}$ is the complete bipartite graph with parts of sizes $s$ and $t$. We determine $\mathrm{wsat}(n, K_{2, t})$, correcting a previous report in the literature. It is also shown that $\mathrm{wsat}(s+t, K_{s,t})=\binom{s+t-1}{2}$ if $\gcd(s, t)=1$ and $\mathrm{wsat}(s+t, K_{s,t})=\binom{s+t-1}{2}+1$, otherwise

Weak Saturation Numbers for Sparse Graphs

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Weak Saturation Numbers for Sparse Graphs

For a fixed graph F, a graph G is F-saturated if there is no copy of F in G, but for any edge e ∉ G, there is a copy of F in G + e. The minimum number of edges in an F-saturated graph of order n will be denoted by sat(n, F). A graph G is weakly F-saturated if there is an ordering of the missing edges of G so that if they are added one at a time, each edge added creates a new copy of F. The minimum size of a weakly F-saturated graph G of order n will be denoted by wsat(n, F). The graphs of order n that are weakly F-saturated will be denoted by wSAT(n, F), and those graphs in wSAT(n, F) with wsat(n, F) edges will be denoted by wSAT(n, F). The precise value of wsat(n, T) for many families of sparse graphs, and in particular for many trees, will be determined. More specifically, families of trees for which wsat(n, T) = |T|−2 will be determined. The maximum and minimum values of wsat(n, T) for the class of all trees will be given. Some properties of wsat(n, T) and wSAT(n, T) for trees will be discussed. Keywords: saturated graphs, sparse graphs, weak saturation