FROM STEINER TRIPLE SYSTEMS TO 3-SUN SYSTEMS

Abstract

An $n$-$sun$ is the graph with $2n$ vertices consisting of an $n$-cycle with $n$ pendent edges which form a 1-factor. In this paper we show that the necessary and sufficient conditions for the decomposition of complete tripartite graphs with at least two partite sets having the same size into $3$-suns and give another construction to get a $3$-sun system of order $n$, for $n\equiv 0,1,4,9$ (mod 12). In the construction we metamorphose a Steiner triple system into a $3$-sun system. We then embed a cyclic Steiner triple system of order $n$ into a $3$-sun system of order $2n-1$, for $n\equiv 1$ (mod 6).