Some NP-complete problems for hypergraph degree sequences
Discrete Applied Mathematics
Numerical characterization of n-cube subset partitioning
Discrete Applied Mathematics
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The question of necessary and sufficient conditions for the existence of a simple hypergraph with a given degree sequence is a long-standing open problem. Let @j"m(n) denote the set of all degree sequences of simple hypergraphs on vertex set [n]={1,2,...,n} that have m edges. A simple characterisation of @j"m(n) is defined in terms of its upper and/or lower elements (degree sequences). In the process of finding all upper degree sequences, a number of results have been achieved in this paper. Classes of upper degree sequences with lowest rank (sum of degrees) r"m"i"n and with highest rank r"m"a"x have been found; in the case of r"m"i"n, the unique class of isomorphic hypergraphs has been determined; the case of r"m"a"x leads to the simple uniform hypergraph degree sequence problem. A smaller generating set has been found for @j"m(n). New classes of upper degree sequences have been generated from the known sequences in dimensions less than n.