Probabilistic Bounds and Domination in Random Graphs

Abstract

In Chapter 6, we presented upper bounds on the domination number of a graph in terms of its order 𝑛 and minimum degree δ. For small δ, the best known bounds to date are summarized in Table 6.5 in Chapter 6. Recall that for δ ∈ [3], the bounds given in the table are tight, while for all values of δ ≥ 4, no tight bound on the domination number is yet known. When δ is sufficiently large, optimal bounds on the domination number can be found using the Probabilistic Method. We present several such probabilistic bounds, including one for the total domination number, in this chapter.