The spectrum for 2-perfect bowtie systems (Q1343249)

The authors define a bowtie in \(K_ v\), the complete graph on \(v\) vertices, as a pair of edge disjoint triangles with a common vertex. Consequently, a bowtie system is an edge disjoint decomposition of \(K_ v\) into bowties. If the additional property holds that each bowtie can be replaced by precisely one of its distance 2 graphs in such a manner that the so obtained collection of bowties is again a bowtie system, then the system is called 2-perfect. The authors prove, by construction, that the spectrum for 2-perfect bowtie systems is \(v\equiv 1\) or \(9\pmod {12}\) (possible exceptions are \(v=69\) and 81). They also show that a 2-perfect decomposition of \(K_ v\setminus K_ 3\) into bowties exists iff \(v\equiv 3\) or \(7\pmod {12}\).