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 ≡ 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 ≡ 1 (mod 6).

A reduction of the spectrum problem for odd sun systems and the prime case

“…As far as we know, this problem has been completely settled only when k = 3, 5 [8,10], k = 4, 6, 8 [12], and when k = 10, 14 or 2 4 t ≥ [9]. It is important to notice that, as a consequence of a general result proved in [14], condition (*) is sufficient whenever v is large enough with respect to k. These results seem to suggest the following.…”

“…Proof The existence of 3‐sun systems and 5‐sun systems has been solved in [10] and in [8], respectively. Hence we can suppose and .…”

“…Since the existence of ‐sun systems with has been proved in [10] and in [8], respectively, here we can assume .…”

“…In this paper we study the existence problem for ‐sun systems of (). Clearly, for such a system to exist we must have As far as we know, this problem has been completely settled only when [8,10], [12], and when or [9]. It is important to notice that, as a consequence of a general result proved in [14], condition (*) is sufficient whenever is large enough with respect to .…”

Decompositions of balanced complete bipartite graphs into suns and stars

“…Liang and Guo [10,11] gave the existence spectrum of a k-sun system of order v as k = 3, 4, 5, 6, 8 by using a recursive construction. Recently, Fu et al [6] investigate the problem of the decomposition of complete tripartite graphs into 3-suns and find the necessary and sufficient condition for the existence of a k-sun system of order v in [7,8].…”