A $3$-uniform hypergraph is a generalization of simple graphs where each hyperedge is a subset of vertices of size $3$. 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 is to decide if a $3$-uniform hypergraph exists with a prescribed degree sequence. Such a hypergraph is called a realization. Recently, Deza \emph{et al.} proved that the degree sequence problem for $3$-uniform hypergraphs is NP-complete. Some special cases are easy; however, polynomial algorithms have been known so far only for some very restricted degree sequences. The main result of our research is the following. If all degrees are between $\frac{2n^2}{63}+O(n)$ and $\frac{5n^2}{63}-O(n)$ in a degree sequence $D$, further, the number of vertices is at least $45$, and the degree sum can be divided by $3$, then $D$ has a $3$-uniform hypergraph realization. Our proof is constructive and in fact, it constructs a hypergraph realization in polynomial time for any degree sequence satisfying the properties mentioned above. To our knowledge, this is the first polynomial running time algorithm to construct a $3$-uniform hypergraph realization of a highly irregular and dense degree sequence.

翻译：$3$-一致超图是简单图的一种推广，其中每条超边是大小为$3$的顶点子集。超图中顶点的度是包含该顶点的超边数量。超图的度序列是其顶点度的序列。$3$-一致超图的度序列问题是判断是否存在一个具有给定度序列的$3$-一致超图，这样的超图称为实现。近期，Deza等人证明了$3$-一致超图的度序列问题是NP完全的。某些特殊情形容易处理；然而，迄今为止仅对非常受限的度序列已知多项式时间算法。我们研究的主要结果如下：若度序列$D$中所有度介于$\frac{2n^2}{63}+O(n)$与$\frac{5n^2}{63}-O(n)$之间，且顶点数至少为$45$，且度数和可被$3$整除，则$D$存在一个$3$-一致超图实现。我们的证明是构造性的，实际上，它能对任何满足上述性质的度序列在多项式时间内构造出一个超图实现。据我们所知，这是首个能在多项式运行时间内构造高度不规则且稠密度序列的$3$-一致超图实现的算法。