Abstract
Consider all k-element subsets and ℓ-element subsets (k>ℓ) of an n-element set as vertices of a bipartite graph. Two vertices are adjacent if the corresponding ℓ-element set is a subset of the corresponding k-element set. Let Gk,ℓ denote this graph. The domination number of Gk,1 was exactly determined by Badakhshian, Katona and Tuza. A conjecture was also stated there on the asymptotic value (n tending to infinity) of the domination number of Gk,2. Here we prove the conjecture, determining the asymptotic value of the domination number γ(Gk,2)=k+32(k−1)(k+1)n2+o(n2).
Highlights
IntroductionThe domination number γ(G) of a graph G is the smallest possible size of a dominating set
Let [n] = {1, 2, . . . , n} be the underlying set and [n] k be the family of all k-element subsets of [n]
It is not much misleading to call Gk,l as the graph defined by the k-th and l-th level
Summary
The domination number γ(G) of a graph G is the smallest possible size of a dominating set. Let us remark that this problem can be seen as a two-sided analogue of an old question of Erdos and Hanani [3] They defined the covering number M(n, k, l) to be the minimum size of a family K ⊂.
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