On diagonal cycle systems (Q2760444)

A diagonal cycle \(D_{2n}\) is a \(2n\)-cycle with an extra chord between two vertices of distance \(n\). The topic of the paper is the decomposition of a complete graph \(K_v\) into edge-disjoint diagonal cycles \( D_{2n} \), also called a \( D_{2n}\)-design of order \( v \). The obtained results are summed up as follows: there exists a \( D_{2n} \)-design of order \( v \) for every \( v \equiv 0 \pmod{2n+1}\) and \(n\) odd (except when \(n=3\) and \(v=7\)); for every \( v \equiv 1 \pmod{2n+1}\) and \(n=5\), \(n=9\), or \(n \equiv 3 \pmod{4} \); and for every \( v \equiv 1 \pmod{4n+2} \) and \( n \equiv 1 \pmod{4} \), \( n \geq 13 \). All proofs are based on recursive constructions from small \( D_{2n} \)-designs some of which were found with the help of a computer. The author conjectures the existence of a \( D_{2n} \)-design of order \( v \equiv 1 \pmod{2n+1} \) for all odd \(n\). The existence of a \( D_{2n} \)-design of order \( v \equiv 2n+2 \pmod{4n+2} \) for \( n \geq 13 \) remains open.