Cyclic Hypergraph Degree Sequences

The problem of efficiently characterizing degree sequences of simple hypergraphs is a fundamental long-standing open problem in Graph Theory. Several results are known for restricted versions of this problem. This paper adds to the list of sufficient conditions for a degree sequence to be hypergraphic. This paper proves a combinatorial lemma about cyclically permuting the columns of a binary table with length n binary sequences as rows. We prove that for any set of cyclic permutations acting on its columns, the resulting table has all of its 2 rows distinct. Using this property, we first define a subset cyclic hyper degrees of hypergraphic sequences and show that they admit a polynomial time recognition algorithm. Next, we prove that there are at least 2 (n−1)(n−2) 2 cyclic hyper degrees, which also serves as a lower bound on the number of hypergraphic sequences. The cyclic hyper degrees also enjoy a structural characterization, they are the integral points contained in the union of some n-dimensional rectangles. 1998 ACM Subject Classification G.2.1 Combinatorics, G.2.2 Hypergraphs