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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Lee codes have been intensively studied for more than 40 years. Interest in these codes has been triggered by the Golomb-Welch conjecture on the existence of the perfect error-correcting Lee codes. In this paper we deal with the existence and enumeration of diameter perfect Lee codes. As main results we determine all $q$ for which there exists a linear diameter-4 perfect Lee code of word length $n$ over $Z_{q},$ and prove that for each $n\geq 3$ there are uncountable many diameter-4 perfect Lee codes of word length $n$ over $Z.$ This is in a strict contrast with perfect error-correcting Lee codes of word length $n$ over $Z\,$\ as there is a unique such code for $n=3,$ and its is conjectured that this is always the case when $2n+1$ is a prime. We produce diameter perfect Lee codes by an algebraic construction that is based on a group homomorphism. This will allow us to design an efficient algorithm for their decoding. We hope that this construction will turn out to be useful far beyond the scope of this paper

There are numerous application of quasigroups in cryptology. It turns out that quasigroups with the relatively small number of associative triples can be utilized in designs of hash functions. In this paper we provide both a new lower bound and a new upper bound on the minimum number of associative triples over quasigroups of a given order.

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