The construction of antipodal triple systems by simulated annealing
A Steiner triple system \(S(2,3,v)\) is a pair \((S,B)\), where \(S\) is a \(v\)-set and \(B\) is a set of 3-subsets of \(S\), called triples, such that any pair of distinct elements of \(S\) is included in exactly one triple. A cyclic triple is a directed 3-cycle. A Mendelsohn triple system \(M(2,3,v)\) is a decomposition of the arcs of the complete digraph (without loops) on \(v\) vertices into cyclic triples. A Steiner triple system \((S,B)\) is called hemispheric if there exists an orientation of \(B\) into a set \(B'\) of cyclic triples and an involution \(f\) of \(S\) having exactly one fixed point such that \(B\cap f(B)=\varnothing\) and such that \((S,B'\cup f(B'))\) is a Mendelsohn triple system. The resulting triple \((S,B,f)\) is then called an antipodal triple system. The authors use simulated annealing on a carefully chosen feasibility space to show that any \(S(2,3,v)\) with \(7\leq v\leq15\) is hemispheric. They conjecture that every \(S(2,3,v)\) with \(v>3\) is hemispheric.
scientific article; zbMATH DE number 932526
scientific article