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- W1575931478 abstract "In this paper we study the behavior of the domination number of the Erdös--Rényi random graph $mathcal{G}(n,p)$. Extending a result of Wieland and Godbole we show that the domination number of $mathcal{G}(n,p)$ is equal to one of two values asymptotically almost surely whenever $p gg frac{ln^2n}{sqrt{n}}$. The explicit values are exactly at the first moment threshold, that is, where the expected number of dominating sets starts to tend to infinity. For small $p$ we also provide various nonconcentration results which indicate why some sort of lower bound on the probability $p$ is necessary in our first theorem. Concentration, though not on a constant length interval, is proven for every $pgg 1/n$. These results show that unlike in the case of $p gg frac{ln^2n}{sqrt{n}}$, where concentration of the domination number happens around the first moment threshold, for $p = O( ln n/n)$ it does so around the median. In particular, in this range the two are far apart from each other." @default.
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- W1575931478 date "2015-01-01" @default.
- W1575931478 modified "2023-10-17" @default.
- W1575931478 title "On the Concentration of the Domination Number of the Random Graph" @default.
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- W1575931478 doi "https://doi.org/10.1137/12090054x" @default.
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