Fast construction on a restricted budget
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866916493277528064 |
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| author | Frieze, Alan Krivelevich, Michael Michaeli, Peleg |
| author_facet | Frieze, Alan Krivelevich, Michael Michaeli, Peleg |
| contents | We introduce a model of a controlled random graph process. In this model, the edges of the complete graph $K_n$ are ordered randomly and then revealed, one by one, to a player called Builder. He must decide, immediately and irrevocably, whether to purchase each observed edge. The observation time is bounded by parameter $t$, and the total budget of purchased edges is bounded by parameter $b$. Builder's goal is to devise a strategy that, with high probability, allows him to construct a graph of purchased edges possessing a target graph property $\mathcal{P}$, all within the limitations of observation time and total budget. We show the following: (a) Builder has a strategy to achieve $k$-vertex-connectivity at the hitting time for this property by purchasing at most $c_kn$ edges for an explicit $c_k<k$; and a strategy to achieve minimum degree $k$ (slightly) after the threshold for minimum degree $k$ by purchasing at most $(1+\varepsilon)kn/2$ edges (which is optimal). (b) Builder has a strategy to create a Hamilton cycle at the hitting time for Hamiltonicity by purchasing at most $Cn$ edges for an absolute constant $C>1$; this is optimal in the sense that $C$ cannot be arbitrarily close to $1$. This substantially extends the classical hitting time result for Hamiltonicity due to Ajtai--Komlós--Szemerédi and Bollobás. (c) Builder has a strategy to create a perfect matching by time $(1+\varepsilon)n\log{n}/2$ while purchasing at most $(1+\varepsilon)n/2$ edges (which is optimal). (d) Builder has a strategy to create a copy of a given $k$-vertex tree if $t\ge b\gg\max\{(n/t)^{k-2},1\}$, and this is optimal; (e) For $\ell=2k+1$ or $\ell=2k+2$, Builder has a strategy to create a copy of a cycle of length $\ell$ if $b\gg\max \{n^{k+2}/t^{k+1},n/\sqrt{t}\}$, and this is optimal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_07251 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Fast construction on a restricted budget Frieze, Alan Krivelevich, Michael Michaeli, Peleg Combinatorics 05C80 (Primary) 60C05, 05C45 (Seconday) We introduce a model of a controlled random graph process. In this model, the edges of the complete graph $K_n$ are ordered randomly and then revealed, one by one, to a player called Builder. He must decide, immediately and irrevocably, whether to purchase each observed edge. The observation time is bounded by parameter $t$, and the total budget of purchased edges is bounded by parameter $b$. Builder's goal is to devise a strategy that, with high probability, allows him to construct a graph of purchased edges possessing a target graph property $\mathcal{P}$, all within the limitations of observation time and total budget. We show the following: (a) Builder has a strategy to achieve $k$-vertex-connectivity at the hitting time for this property by purchasing at most $c_kn$ edges for an explicit $c_k<k$; and a strategy to achieve minimum degree $k$ (slightly) after the threshold for minimum degree $k$ by purchasing at most $(1+\varepsilon)kn/2$ edges (which is optimal). (b) Builder has a strategy to create a Hamilton cycle at the hitting time for Hamiltonicity by purchasing at most $Cn$ edges for an absolute constant $C>1$; this is optimal in the sense that $C$ cannot be arbitrarily close to $1$. This substantially extends the classical hitting time result for Hamiltonicity due to Ajtai--Komlós--Szemerédi and Bollobás. (c) Builder has a strategy to create a perfect matching by time $(1+\varepsilon)n\log{n}/2$ while purchasing at most $(1+\varepsilon)n/2$ edges (which is optimal). (d) Builder has a strategy to create a copy of a given $k$-vertex tree if $t\ge b\gg\max\{(n/t)^{k-2},1\}$, and this is optimal; (e) For $\ell=2k+1$ or $\ell=2k+2$, Builder has a strategy to create a copy of a cycle of length $\ell$ if $b\gg\max \{n^{k+2}/t^{k+1},n/\sqrt{t}\}$, and this is optimal. |
| title | Fast construction on a restricted budget |
| topic | Combinatorics 05C80 (Primary) 60C05, 05C45 (Seconday) |
| url | https://arxiv.org/abs/2207.07251 |