A Note on the Existence of Cyclic and \(1\)-Rotational Kite Systems

Abstract

A kite graph is a graph obtained from a \(3\)-cycle (or triple) by adding a pendent edge to a vertex of the \(3\)-cycle. A kite system of order \(v\) is a pair \((X, \mathcal{B})\), where \(\mathcal{B}\) is an edge-disjoint collection of kite graphs which partitions the edge set of \(K_v\). A kite system of order \(v\) is cyclic if it admits an automorphism of order \(v\), and 1-rotational if it admits an automorphism containing one fixed point and a cycle of length \(v – 1\). In this paper, we show that there exists a cyclic kite system of order \(v\) if and only if \(v \equiv 1 \pmod{8}\), and there exists a \(1\)-rotational kite system of order \(v\) if and only if \(v \equiv 0 \pmod{8}\).