On the number of precolouring extensions (Q1590211)

A precolouring of a hypergraph \(H\) is a partial mapping of the vertex set of \(H\) into the natural numbers. A colouring of \(H\) is a precolouring that is defined on the entire vertex set of \(H\). A colouring or precolouring is proper if it induces no monochromatic edges apart from loops. The author proves that the number of proper \(\lambda\)-colourings of a hypergraph extending a given proper precolouring agrees with a polynomial \(\lambda\) for any sufficiently large \(\lambda\). He then establishes a generalization of Whitney's broken circuit theorem [see \textit{H. Whitney}, Bull. Am. Math. Soc. 38, 572-579 (1932; Zbl 0005.14602)], by using an improved version of the inclusion-exclusion principle.