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- W4285787411 abstract "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--Koml'os--Szemer'edi and Bollob'as. (c) Builder has a strategy to create a perfect matching by time $(1+varepsilon)nlog{n}/2$ 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 $tge bggmax{(n/t)^{k-2},1}$, 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 $bggmax {n^{k+2}/t^{k+1},n/sqrt{t}}$, and this is optimal." @default.
- W4285787411 created "2022-07-19" @default.
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- W4285787411 date "2022-07-14" @default.
- W4285787411 modified "2023-09-25" @default.
- W4285787411 title "Fast construction on a restricted budget" @default.
- W4285787411 doi "https://doi.org/10.48550/arxiv.2207.07251" @default.
- W4285787411 hasPublicationYear "2022" @default.
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