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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
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References:
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