On the Concentration of the Domination Number of the Random Graph

Abstract

Abstract. In this paper we study the behaviour of the domination number of the Erdős-Rényi random graph G(n, p). Extending a result of Wieland and Godbole we show that the domination number of G(n, p) is equal to one of two values asymptotically almost surely whenever p ≫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 non-concentration 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 p ≫ 1/n. These results show that unlike in the case ofwhere 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.

The Bondage Number of Random Graphs

“…It is known that even for sparser graphs (namely, for p = p n ≫ log 2 n/ √ n, but bounded away from 1) a.a.s. the domination number of G (n, p) takes one out of two consecutive integer values, r or r + 1, where r = r n is defined in (4) (see [4] and also [8] for an earlier paper where denser graphs were considered). The next result shows that if f (n, r, p) (that is, the expected number of dominating sets of cardinality r) is large, then we actually have one-point concentration and the bondage number is of order pn.…”

Tropical dominating sets in vertex-coloured graphs

“…Recently Glebov, Liebenau and Szabó [21] strengthened this two-point concentration result by extending the range of p down to (log 2 n)/ √ n . We are interested here in the maximum number of colors c that can be used so that a.a.s.…”

(Total) Domination in Prisms

“…With respect to the conjecture we refer to [5] for the investigation of the behavior of the domination number in random graphs.…”