Hexagon quadrangle systems (Q2468002)

Let \(\lambda K_n\) be the complete graph with vertex set \(X\) where every pair of vertices is joined by \(\lambda\) edges. A \(\lambda\)-fold \(m\)-cycle system of order \(n\) is a pair \((X,\mathcal{C})\), where \(X\) is a finite set of \(n\) elements, called vertices, and \(\mathcal{C}\) is a collection of disjoint \(m\)-cycles which partitions the edge set of \(\lambda K_n\). A hexagonal quadrangle is the graph obtained from a 6-cycle \((X_1,X_2,X_3,X_4,X_5,X_6)\) by adding the chords \(\{ X_1,X_3 \}\) and \(\{ X_4,X_6 \}\). A hexagonal quadrangle system of order \(n\) and index \(\rho\) is a pair \((X,H)\) where \(X\) is a finite set of \(n\) vertices and \(H\) is a collection of edge disjoint hexagon quadrangles (called blocks) which partitions the edge set of \(\rho K_n\), with vertex set \(X\). A hexagon quadrangle system is said to be a 4-nesting if the collection of the 4-cycles contained in the hexagon quadrangles is a \(\frac{\rho}{2}\)-fold 4-cycle system. It is said to be a 6-nesting if the collection of 6-cycles contained in the hexagon quadrangles is a \(\frac{3 \rho}{4}\)-fold 6-cycle system. It is said to be a \((4,6)\)-nesting, briefly a \(N(4,6)\)-HQS, if it is both a 4-nesting and a 6-nesting. In the paper under review, the author completely determines the spectrum of \(N(4,6)\)-HQS for \(\lambda = 6h\), \(\mu=4h\) and \(\rho=8h\), \(h\) positive integer.