Colorings of (r, r)-Uniform, Complete, Circular, Mixed Hypergraphs

In colorings of some block designs, the vertices of blocks can be thought of as hyperedges of a hypergraph 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 H is called circular if there exists a host cycle on the vertex set X such that every edge (C- or 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 χ(H)=2 and χ¯(H)=n−s or n−s−1 where s is the sieve number.