On $3$-Hypergraphs with Equal Degree Sequences

Abstract

The question of necessary and sufficient conditions for the existence of a simple $3$-uniform hypergraph with a given degree sequence is a long outstanding open question. We provide a result on degree sequences of $3$-hypergraphs which shows that any two $3$-hypergraphs with the same degree sequence can be transformed into each other using a sequence of trades, also known as null-$3$-hypergraphs. This result is similar to the Havel-Hakimi theorem for degree sequences of graphs.