**MSC:**- 05B15 Orthogonal arrays, Latin squares, Room squares
- 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 two-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 two-colorable ODC of $K_q$ by a Hamiltonian path, then there is an

ODC of $K_{qn}$ by a Hamiltonian path. In \cite{ul00}, two-colorable ODCs of

$K_n$ and $K_{2n}$, respectively, by Hamiltonian paths were constructed

for all odd square numbers $n\ge 9$. Here we continue this work and construct

cyclic two-colorable ODCs of $K_n$ and $K_{2n}$, respectively, by Hamiltonian

paths for all $n$ of the form $n=4k^2+1$ or $n=(k^2+1)/2$ with some

integer $k$.