The spectrum for 2-perfect bowtie systems Original Research Article

A bowtie is a pair of edge disjoint triangles of Kv with a common vertex. A bowtie system is an edge disjoint decomposition of Kv into bowties. A bowtie system is 2-perfect if it has the additional property that each bowtie can be replaced by exactly one of its distance 2 graphs so that the resulting collection of bowties is also a bowtie system. We show that the spectrum of 2-perfect bowtie systems is precisely the set of all n1 or 9 (mod 12), with the possible exceptions of n=69 and 81. We also solve the same problem for Kv⧹K3. That is, we show that a 2-perfect decomposition of Kv⧹K3 into bowties exists if and only if v3 or 7 (mod 12).