Saturation numbers of bipartite graphs in random graphs

Abstract: For a given graph $F$, the $F$-saturation number of a graph $G$, denoted by ${sat}(G, F)$, is the minimum number of edges in an edge-maximal $F$-free subgraph of $G$. In 2017, Kor\'andi and Sudakov determined ${sat}({G}(n, p), K_r)$ asymptotically, where ${G}(n, p)$ denotes the Erd\H{o}s-R\'enyi random graph and $K_r$ is the complete graph on $r$ vertices. In this paper, among other results, we present an asymptotic upper bound on ${sat}({G}(n, p), F)$ for any bipartite graph $F$ and also an asymptotic lower bound on ${sat}({G}(n, p), F)$ for any complete bipartite graph $F$.