Semester Report SS 04 of Daniela Kühn
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چکیده
A classical theorem of Dirac states that every graph on n vertices with minimum degree at least n/2 contains a Hamilton cycle. If one seeks an analogue of this result for uniform hypergraphs, then several alternatives suggest themselves. In the following, we will restrict ourselves to 3-uniform hypergraphs H. Thus each hyperedge of H consists of precisely 3 vertices. A natural way to extend the notion of the minimum degree of a graph to that of a 3-uniform hypergraph H is the following. Given two distinct vertices x and y of H, the neighbourhood N(x, y) of x and y in H is the set of all those vertices z which form a hyperedge together with x and y. The minimum degree δ(H) is defined to be the minimum |N(x, y)| over all pairs of vertices of H. A 3-uniform hypergraph C is called a tight cycle if C has at least 4 vertices and if there is a cyclic ordering of the vertices of C such that every 3 consecutive vertices form a hyperedge of C and C contains no other hyperedges. Note that the cyclic ordering of the vertices of C induces a cyclic ordering of the hyperedges of C.
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تاریخ انتشار 2003