**MSC:**- 05C70 Factorization, matching, covering and packing
- 05B40 Packing and covering, See also {11H31, 52C15, 52C17}

K_n by an almost hamiltonian cycle is a decomposition of

2K_n into cycles of length n-1 such that the intersection

of any two of them is exactly one edge.

We introduce a new class of such decompositions.

If n is a prime, the special structure of such a

decomposition allows to expand it to an ODC of K_{n+1} by

an almost hamiltonian cycle. This yields the existence of

an ODC of K_{p+1} by an almost hamiltonain cycle for

primes p of order 3 mod 4 and its eventual existence for

arbitrary primes p.