| Summary: | In colorings of some block designs, the vertices of blocks can be thought of as hyperedges of a hypergraph <inline-formula><math display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula> 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 <inline-formula><math display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula> is called circular if there exists a host cycle on the vertex set <i>X</i> such that every edge (<inline-formula><math display="inline"><semantics><mi mathvariant="script">C</mi></semantics></math></inline-formula>- or <inline-formula><math display="inline"><semantics><mi mathvariant="script">D</mi></semantics></math></inline-formula>-) induces a connected subgraph of this cycle. We propose an algorithm to color the <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>r</mi><mo>,</mo><mi>r</mi><mo>)</mo></mrow></semantics></math></inline-formula>-uniform, complete, circular, mixed hypergraphs for all feasible values with no gaps. In doing so, we show <inline-formula><math display="inline"><semantics><mrow><mi>χ</mi><mo>(</mo><mi mathvariant="script">H</mi><mo>)</mo><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mover accent="true"><mi>χ</mi><mo>¯</mo></mover></semantics></math></inline-formula><inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="script">H</mi><mo>)</mo><mo>=</mo><mi>n</mi><mo>−</mo><mi>s</mi></mrow></semantics></math></inline-formula> or <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>−</mo><mi>s</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> where <i>s</i> is the sieve number.
|
|---|