We consider a problem formulated by Marco Buratti concerning Hamiltonian paths in the complete graph on $Z_p$, $p$ an odd prime.

In this paper, we show that the edge set of a cubic graph can always be partitioned into 10 subsets, each of which induces a matching in the graph. This result is a special case of a general conjecture made by Erdos and NeSetiil: For each d 2 3, the edge set of a graph of maximum degree d can always be partitioned into [5d2/4] subsets each of which induces a matching. 0 1993 John Wiley & Sons, Inc. INTRODUCTIONThroughout this paper, we consider colorings of the edges of a graph with positive integers. Formally, a t-coloring of a graph G = ( V , E ) is a map $: -{1,2,. . . , t}. A t-coloring is proper if $(e) = $(f) and e # f imply that the edges e and f have no common endpoints. Of course, the chromatic index of a graph G is the least t for which G has a proper tcoloring. Note that whenever qj is a proper t-coloring of a graph G = ( V , E ) and a E {1,2,. . . , t}, then the edges in 34 = {e E E:$(e) = a } form a matching in G.An induced matching 34 in a graph G = ( V , E ) is a matching such that no two edges of 34 are joined by an edge of G. In other words, an induced matching is an induced subgraph in which every vertex has degree one. A

We discuss a special case of the Hamilton-Waterloo problem in which a 2-factorization of Kn is sought consisting of 2-factors of two kinds: Hamiltonian cycles, and triangle-factors. We determine completely the spectrum of solutions for several inÿnite classes of orders n.

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