A note on the decomposition of graphs into isomorphic matchings

All graphs considered are finite, undirected, with no loops and no multiple edges. A graph H is said to have a G-decomposition if it is the union of pairwise edge-disjoint subgraphs each isomorphic to G. We denote this situation by GIH. Many results are known about G-decomposition, for references see e.g. [1] and [6]. In this paper we establish some necessary and sufficient conditions for a graph H to have a tK2-decomposition, where tK2 is the graph consisting of t independent edges. Our result implies, as a very special case, the main result of Bialostocki and Roditty [3], that states that if G is a graph with e edges and maximum degree A, then, with a finite number of exceptions, 3K2IG iff 3le and A<=e/3. For every graph G, E(G) is the set of edges of G and e(G)= IE(G)I. //(63 is the maximum degree of G and z'(G) is the chromatic index (=edge-chromatic number) of G. We begin with the following simple lemma, which is proved in [2]: