On the Chromatic Number of the Products of Hypergraphs

Abstract

This paper discusses the chromatic number of the products of $n+1$ -chromatic hypergraphs. The following two results are proved:
Suppose $G$ and $H$ are $n+1$ -chromatic hypergraphs such that each of $G$ and $H$ contains a complete sub-hypergraph of order n and each of $G$    and $H$ contains a vertex critical $n+1$-chromatic sub-hypergraph which has non-empty intersection with the corresponding complete sub-hypergraph of order $n$. Then the product $G\times H$is of chromatic number $n+1$.
Suppose $G$ is an $n+1$-chromatic hypergraph such that each vertex of $G$ is contained in a complete sub-hypergraph of order n. Then for any $n+1$-chromatic hypergraph $H$, $G\times H$ is an $n+1$-chromatic hypergraph.