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A contribution to the chromatic theory of uniform hypergraphs. (English) Zbl 0831.05029
Let $$H$$ be a hypergraph and $$P_H (\lambda)$$ denote the number of proper $$\lambda$$-colourings of $$H$$, where ‘proper’ means that the only monochromatic edges are loops. For uniform hypergraphs $$H$$, lower bounds and upper bounds on $$P_H (\lambda)$$ are obtained in terms of the rank, the number of vertices and edges and some arbitrary number less than the girth. If $$H$$ has $$m > 1$$ edges and is uniform of rank $$r > 1$$, it follows that $$\chi (H) \leq \lceil \root {r - 1} \of m \rceil$$, where $$\chi (H)$$ denotes the chromatic number of $$H$$. It should be noted that the bounds on $$P_H (\lambda)$$ are new for graphs, too.
Reviewer: K.Dohmen (Berlin)

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C65 Hypergraphs
##### Keywords:
hypergraph; bounds; chromatic number
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##### References:
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