Dense, irregular, yet always-graphic 3-uniform hypergraph degree sequences

Abstract

A 3-uniform hypergraph is a generalization of a simple graph where each hyperedge is a subset of exactly three vertices. The degree of a vertex in a hypergraph is the number of hyperedges incident with it. The degree sequence of a hypergraph is the sequence of the degrees of its vertices. The degree sequence problem for 3-uniform hypergraphs asks whether a 3-uniform hypergraph with a given degree sequence exists. Such a hypergraph is called a realization. Recently, Deza et al. proved that this problem is NP-complete. Although some special cases are simple, polynomial-time algorithms are only known for highly restricted degree sequences. The main result of our research is the following: if all degrees in a sequence D of length n are between $\frac{2n^2}{63}+O\left(n\right)$ and $\frac{5n^2}{63}-O\left(n\right)$, the number of vertices is at least 45, and the degree sum is divisible by 3, then D has a 3-uniform hypergraph realization. Our proof is constructive, providing a polynomial-time algorithm for constructing such a hypergraph. To our knowledge, this is the first polynomial-time algorithm to construct a 3-uniform hypergraph realization of a highly irregular and dense degree sequence.