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arXiv · 2304.07731

Saturation numbers of bipartite graphs in random graphs

Meysam Miralaei·Ali Mohammadian·Behruz Tayfeh-Rezaie·Maksim Zhukovskii
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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 $F$.

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Meysam Miralaei, Ali Mohammadian, Behruz Tayfeh-Rezaie, Maksim Zhukovskii. 2023-04-16. Saturation numbers of bipartite graphs in random graphs. https://arxiv.org/abs/2304.07731

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