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

graph $G$ is a collection $\mathcal{G}$ of $n$ spanning subgraphs of $K_n$,

all isomorphic to $G$, such that any two members of $\mathcal{G}$ share

exactly one edge and every edge of $K_n$ is contained in exactly two

members of $\mathcal{G}$. In the 1980's Hering posed the problem to decide

the existence of an ODC for the case that $G$ is an almost-hamiltonian

cycle, i.e.~a cycle of length $n-1$. It is known that the existence of

an ODC of $K_n$ by a hamiltonian path implies the existence of ODCs of

$K_{4n}$ and $K_{16n}$, respectively, by almost-hamiltonian cycles.

Horton and Nonay introduced 2-colorable ODCs and showed: If for $n\ge 3$

and a prime power $q\ge 5$ there are an ODC of $K_n$ by a hamiltonian path

and a 2-colorable ODC of $K_q$ by a hamiltonian path, then there is an

ODC of $K_{qn}$ by a hamiltonian path. We construct 2-colorable ODCs of

$K_n$ and $K_{2n}$, respectively, by hamiltonian paths for all odd square

numbers $n\ge 9$.