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 发表日期：2017年6月23日

The graph-function P(G; \lambda) for graphs was extended to the case for hypergraphs in 1970s. A (simple) hypergraph \mathcal{H}=(\mathcal{V}, \mathcal{E}) consists of a vertex-set \mathcal{V}= \{v_1,...,v_n\} and an edge-set \mathcal{E} =\{e_1,...,e_m\} where e_i\subseteq \mathcal{V} and e_i\neq e_j for all 1\le i\le j. If |e_i|= 2 holds for each e_i\in\mathcal{E} , then \mathcal{H} is a simple graph. A (weak) proper\lambda-colouring of a hypergraph \mathcal{H} is a mapping f : \mathcal{V}\rightarrow \{1,...,\lambda\} such that |\{f(u) : u \in e\}|\ge 2 holds for each e\in \mathcal{E}. Let P(\mathcal{H}; \lambda) be the number of perper \lambda-colourings of \mathcal{H}. This graph-function P(\mathcal{H}; \lambda) is a polynomial of \lambda with degree |\mathcal{V}| and is called the chromatic polynomial of \mathcal{H}. Clearly P(\mathcal{H}; \lambda) is an extension of chromatic polynomials of simple graphs. In this talk, we will present our latest results on some properties of P(H; ) which are different from properties of chromatic polynomials of graphs. We will also introduce some open problems on the study of P(\mathcal{H}; \lambda).

Joint work with Zhang Ruixue, Nanyang Technological University, Singapore.

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