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

Nicholas Newman,Vitaly Voloshin

doi:10.3390/math9080828

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.

Highlights

We determine the range of color classes. It follows that these colorings can be modeled by circular mixed hypergraphs
We introduce a coloring algorithm that will properly color the class of mixed hypergraphs
We focus on a particular family of hypergraphs, the (r, r )-uniform complete circular hypergraphs

Summary

Introduction

Colorings of (r, r)-Uniform, Complete, Circular, Mixed Hypergraphs. As with the colorings of our family of graphs, each e-star could be considered a block and one would want to avoid a monochromatic coloring of any block. We determine the range of color classes It follows that these colorings can be modeled by circular mixed hypergraphs. In the traditional theory of coloring graphs and hypergraphs [5,6,7], we seek colorings of the vertices so that each edge has at least two vertices of different colors. One can seek the dual question to color the vertices so that each edge requires at least two vertices of the same color and ask for the maximum number of colors needed. We introduce a coloring algorithm that will properly color the class of mixed hypergraphs.

Preliminaries

Coloring Algorithm

Conclusions