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Consider all $k$-element subsets and $\ell$-element subsets $(k>\ell )$ of an$n$-element set as vertices of a bipartite graph. Two vertices are adjacent ifthe corresponding $\ell$-element set is a subset of the corresponding$k$-element set. Let $G_{k,\ell}$ denote this graph. The domination number of$G_{k,1}$ was exactly determined by Badakhshian, Katona and Tuza. A conjecturewas also stated there on the asymptotic value ($n$ tending to infinity) of thedomination number of $G_{k,2}$. Here we prove the conjecture, determining theasymptotic value of the domination number $\gamma (G_{k,2})={k+3\over2(k-1)(k+1)}n^2+o(n^2)$.

The domination number of the graph defined by two levels of the $n$-cube, II