Abstract
The concept of a matroid is generalized by imitating the generalization of a graph to a hypergraph.
Two basic examples of such hypermatroids are given. The first type models the coloring problem for hypergraphs whereas the second one is related to coverings of hypergraphs. This second example, covering hypermatroids, is the proper extension of trivial matroids. Covering hypermatroids are characterized by a generalized semimodularity condition.
It is shown how all hypermatroids can be naturally constructed from suitable matroids.
A Poincaré polynomial is defined for hypermatroids. The coloring polynomial for hypergraphs is seen to be a special case. For covering hypermatroids the Poincaré polynomial yields a polynomial identity which gives an enumerative relationship between the covering sets and the transversal sets of a hypergraph.
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© 1974 Springer-Verlag
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Helgason, T. (1974). Aspects of the theory of hypermatroids. In: Berge, C., Ray-Chaudhuri, D. (eds) Hypergraph Seminar. Lecture Notes in Mathematics, vol 411. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0066195
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DOI: https://doi.org/10.1007/BFb0066195
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