In this paper, we investigate partitions of highly symmetrical discrete structures called cycloids. In general, a mixed hypergraph has two types of hyperedges. The vertices are colored in such a way that each C-edge has two vertices of the same color, and each D-edge has two vertices of distinct colors. In our case, a mixed cycloid is a mixed hypergraph whose vertices can be arranged in a cyclic order, and every consecutive p vertices form a C-edge, and every consecutive q vertices form a D-edge in the ordering. We completely determine the maximum number of colors that can be used for any $p\ge 3$ and any $q\ge 2$. We also develop an algorithm that generates a coloring with any number of colors between the minimum and maximum. Finally, we discuss the colorings of mixed cycloids when the maximum number of colors coincides with its upper bound, which is the largest cardinality of a set of vertices containing no C-edge.

In colorings of some block designs, the vertices of blocks can be thought of as hyperedges of a hypergraph $\mathcal{H}$ that can be placed on a circle and colored according to some rules that are related to colorings of circular mixed hypergraphs. A mixed hypergraph $\mathcal{H}$ is called circular if there exists a host cycle on the vertex set X such that every edge ($\mathcal{C}$- or $\mathcal{D}$-) induces a connected subgraph of this cycle. We propose an algorithm to color the $(r,r)$-uniform, complete, circular, mixed hypergraphs for all feasible values with no gaps. In doing so, we show $\chi \left(\mathcal{H}\right)=2$ and $\overline{\chi }$$\left(\mathcal{H}\right)=n-s$ or $n-s-1$ where s is the sieve number.