On antichain intersection numbers, total clique covers and regular graphs (Q1322237)

Let \({\mathfrak F}= \{S_ 1,S_ 2,\dots,S_ p\}\) be a family of distinct nonempty sets. The intersection graph of \(\mathfrak F\) is denoted \(\Omega({\mathfrak F})\). The intersection number \(\omega(G)\) of a graph \(G\) is the minimum cardinality of the union of \(\mathfrak F\) such that \(G\) is the intersection graph of \(\mathfrak F\). The numbers \(\omega_{ai}\) and \(\omega_ m\) corresponds to the case when \(\mathfrak F\) is an antichain or multifamily, respectively. The author determines the numbers \(\omega_{ai}(G)\) in terms of the total clique covering of \(G\). For regular graphs with degree up to 5 and some special graphs the numbers \(\omega_{ai}(G)\) and \(\omega_ m(G)\) are found.