Weak saturation numbers of large complete bipartite graphs
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918131262291968 |
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| author | Akhmejanova, Margarita Vorobyev, Ilya Zhukovskii, Maksim |
| author_facet | Akhmejanova, Margarita Vorobyev, Ilya Zhukovskii, Maksim |
| contents | An $n$-vertex graph $G$ is weakly $F$-saturated if $G$ contains no copy of $F$ and there exists an ordering of all edges in $E(K_n) \setminus E(G)$ such that, when added one at a time, each edge creates a new copy of $F$. The minimum size of a weakly $F$-saturated graph $G$ is called the weak saturation number $\mathrm{wsat}(n, F)$. We obtain exact values and new bounds for $\mathrm{wsat}(n, K_{s,t})$ in the previously unaddressed range $s+t < n < 3t-3$, where $3\leq s\leq t$. To prove lower bounds, we introduce a new method that takes into account connectivity properties of subgraphs of a complement $G'$ to a weakly saturated graph $G$. We construct an auxiliary hypergraph and show that a linear combination of its parameters always increases in the process of the deletion of edges of $G'$. This gives a lower bound which is tight, up to an additive constant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_19435 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weak saturation numbers of large complete bipartite graphs Akhmejanova, Margarita Vorobyev, Ilya Zhukovskii, Maksim Combinatorics An $n$-vertex graph $G$ is weakly $F$-saturated if $G$ contains no copy of $F$ and there exists an ordering of all edges in $E(K_n) \setminus E(G)$ such that, when added one at a time, each edge creates a new copy of $F$. The minimum size of a weakly $F$-saturated graph $G$ is called the weak saturation number $\mathrm{wsat}(n, F)$. We obtain exact values and new bounds for $\mathrm{wsat}(n, K_{s,t})$ in the previously unaddressed range $s+t < n < 3t-3$, where $3\leq s\leq t$. To prove lower bounds, we introduce a new method that takes into account connectivity properties of subgraphs of a complement $G'$ to a weakly saturated graph $G$. We construct an auxiliary hypergraph and show that a linear combination of its parameters always increases in the process of the deletion of edges of $G'$. This gives a lower bound which is tight, up to an additive constant. |
| title | Weak saturation numbers of large complete bipartite graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2508.19435 |