Weak saturation numbers in random graphs
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929268960788480 |
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| author | Kalinichenko, Olga Miralaei, Meysam Mohammadian, Ali Tayfeh-Rezaie, Behruz |
| author_facet | Kalinichenko, Olga Miralaei, Meysam Mohammadian, Ali Tayfeh-Rezaie, Behruz |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_10375 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Weak saturation numbers in random graphs Kalinichenko, Olga Miralaei, Meysam Mohammadian, Ali Tayfeh-Rezaie, Behruz Combinatorics 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. |
| title | Weak saturation numbers in random graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2306.10375 |