Nonconvexity of the set of hypergraph degree sequences

TL;DR

This paper demonstrates that the set of degree sequences for certain hypergraphs is nonconvex, contrasting with the well-understood convex structure of graph degree sequences, revealing fundamental differences in hypergraph degree sequence sets.

Contribution

It proves the nonconvexity of hypergraph degree sequence sets for uniform hypergraphs with specific parameters, extending known graph results to hypergraphs.

Findings

Hypergraph degree sequence sets are not convex for k ≥ 3 and n ≥ k+13.

Nonconvexity also holds for k-partite and λ-balanced hypergraphs.

Contrasts with the convex nature of graph degree sequence sets.

Abstract

It is well known that the set of possible degree sequences for a graph on $n$ vertices is the intersection of a lattice and a convex polytope. We show that the set of possible degree sequences for a $k$-uniform hypergraph on $n$ vertices is not the intersection of a lattice and a convex polytope for $k \geq 3$ and $n \geq k+13$. We also show an analogous nonconvexity result for the set of degree sequences of $k$-partite $k$-uniform hypergraphs and the generalized notion of $\lambda$-balanced $k$-uniform hypergraphs.