# Cyclic Hypergraph Degree Sequences

## TL;DR

This paper introduces cyclic hyper degrees, a new class of hypergraph degree sequences, providing a polynomial recognition algorithm and establishing a lower bound on their abundance, advancing understanding of hypergraph degree characterization.

## Contribution

It defines cyclic hyper degrees, proves their recognition is polynomial, and establishes a lower bound on their quantity, contributing new structural insights into hypergraph degree sequences.

## Key findings

- Cyclic hyper degrees admit a polynomial time recognition algorithm.
- There are at least 2^{(n-1)(n-2)/2} cyclic hyper degrees.
- Cyclic hyper degrees are characterized as integral points in unions of n-dimensional rectangles.

## Abstract

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 {\em 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 $2^n$ rows distinct. Using this property, we first define a subset {\em cyclic hyper degrees} of hypergraphic sequences and show that they admit a polynomial time recognition algorithm. Next, we prove that there are at least $2^{\frac{(n-1)(n-2)}{2}}$ {\em cyclic hyper degrees}, which also serves as a lower bound on the number of {\em hypergraphic} sequences. The {\em cyclic hyper degrees} also enjoy a structural characterization, they are the integral points contained in the union of some $n$-dimensional rectangles.