No 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).

The originally published article incorrectly listed the university name as John's University.

The originally published article incorrectly listed the university name as John's University.

In this paper, we extend the work on minimum coverings of Kn with triangles. We prove that when P is any forest on n vertices the multigraph G = Kn ∪ P can be decomposed into triangles if and only if three trivial necessary conditions are satisÿed: (i) each vertex in G has even degree, (ii) each vertex in P has odd degree, and (iii) the number of edges in G is a multiple of 3. This result is of particular interest because the corresponding packing problem where the leave is any forest is yet to be solved. We also consider some other families of packings, and provide a variation on a proof by Colbourn and Rosa which settled the packing problem when P is any 2-regular graph on at most n vertices.