论文标题
完美的超图颜色
Perfect colorings of hypergraphs
论文作者
论文摘要
对图的完美色彩（公平分区）进行了广泛的研究，而相同的HyperGraphs概念吸引了大量的关注。本文的目的是为超图制定完美色彩的基本概念和特性。首先，我们引入了一个多维矩阵方程，以完美的超图颜色，并将此定义与基于入射图的标准方法进行比较。接下来，我们表明完美着色的参数矩阵的特征值是超图的多维邻接矩阵的特征值。我们认为超图的覆盖物是一种完美着色的特殊情况，并证明了存在两个超图的常见覆盖物的定理。例如，我们表明，超图中的$ k $ transversal对应于完美的着色并计算其参数。最后，我们发现了Fano飞机超图的所有完美$ 2 $颜色，并计算了此超图的某些特征值。
Perfect colorings (equitable partitions) of graphs are extensively studied, while the same concept for hypergraphs attracts much less attention. The aim of this paper is to develop basic notions and properties of perfect colorings for hypergraphs. Firstly, we introduce a multidimensional matrix equation for perfect colorings of hypergraphs and compare this definition with a standard approach based on the incidence graph. Next, we show that the eigenvalues of the parameter matrix of a perfect coloring are eigenvalues of the multidimensional adjacency matrix of a hypergraph. We consider coverings of hypergraphs as a special case of perfect colorings and prove a theorem on the existence of a common covering of two hypergraphs. As an example, we show that a $k$-transversal in a hypergraph corresponds to a perfect coloring and calculate its parameters. At last, we find all perfect $2$-colorings of the Fano's plane hypergraph and compute some eigenvalues of this hypergraph.
