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

Author

Listed:
  • Nicholas Newman

    (Department of Mathematics and Statistics, Troy University, 600 University Ave., Troy, AL 36082, USA)

  • Vitaly Voloshin

    (Department of Mathematics and Statistics, Troy University, 600 University Ave., Troy, AL 36082, USA)

Abstract

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.

Suggested Citation

  • Nicholas Newman & Vitaly Voloshin, 2021. "Colorings of (r, r)-Uniform, Complete, Circular, Mixed Hypergraphs," Mathematics, MDPI, vol. 9(8), pages 1-5, April.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:8:p:828-:d:533557
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