Twofold 2-perfect bowtie systems with an extra property

Abstract

A bowtie is a closed trail whose graph consists of two 3-cycles with exactly one vertex in common. A 2-fold bowtie system of order n is an edge-disjoint decomposition of 2Kn into bowties. A 2-fold bowtie system is said to be 2-perfect provided that every pair of distinct vertices is joined by two paths of length 2. It is said to be extra provided these two paths always have distinct midpoints. The extra property guarantees that the two paths x, a, y and x, b, y between every pair of vertices form a 4-cycle (x, a, y, b), and that the collection of all such 4-cycles is a four-fold 4-cycle system. We show that the spectrum for extra 2-perfect 2-fold bowtie systems is precisely the set of all n ≡ 0 or 1 (mod 3), \({n\,\geqslant\,6}\) . Additionally, with an obvious definition, we show that the spectrum for extra 2-perfect 2-fold maximum packings of 2Kn with bowties is precisely the set of all n ≡ 2 (mod 3), \({n\,\geqslant\,8}\) .