On the Domination Number of a Random Graph

Abstract

In this paper, we show that the domination number $D$ of a random graph enjoys as sharp a concentration as does its chromatic number $\chi$. We first prove this fact for the sequence of graphs $\{G(n,p_n\},\; n\to\infty$, where a two point concentration is obtained with high probability for $p_n=p$ (fixed) or for a sequence $p_n$ that approaches zero sufficiently slowly. We then consider the infinite graph $G({\bf Z}^+, p)$, where $p$ is fixed, and prove a three point concentration for the domination number with probability one. The main results are proved using the second moment method together with the Borel Cantelli lemma.

The Bondage Number of Random Graphs

“…Finally, for r − 3 log log n/p ≤ i ≤ r, by Lemma 7, since Q i ≤ 1, and by Chernoff's bound (see (8)), where the last inequality follows from p ≫ log 2 n/ √ n. Combining all bounds,…”

“…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.…”

“…To do so, we will use several bounds on EW i that are stated in Proposition 8 below. In fact, the variance of X r+1 was already studied in [4] and [8], and we follow some of the ideas in their computations, but we need a more accurate estimation of the error terms involved. Also, the aforementioned papers deal with X r+1 instead of X r , since they make use of the fact that EX r+1 = exp(Θ(log 2 n)).…”

Tropical dominating sets in vertex-coloured graphs

“…The domination number of the random graph G(n, p) has been well studied, see for example [16], [28] and [35]. In Particular, Wieland and Godbole [35] proved the following two-point concentration result.…”

Minimum Degree and Dominating Paths

“…For a > 1/(k + 2) and n sufficiently large, what is the least s such that every k-connected n-vertex graph G with δ(G) ≥ a · n has a dominating path with at most s vertices? (Since G is connected, the value is at most logarithmic in n. If k ∈ O(log n), then the results [8,13] on domination in random graphs imply that s cannot be sublogarithmic, but the question becomes more interesting when k grows faster than log n.) Question 3. For fixed s ∈ N and n sufficiently large, what is the least t such that every n-vertex graph with minimum degree at least t has a dominating path with at most s vertices?…”