Work

A complete dichotomy theorem on the sparse t-uniform hypergraphicality problem

Year: 2026

Type: article

Abstract: We prove a complete dichotomy theorem for the parameterized sparse t -uniform hypergraphical degree sequence problem, sparse - t - uni - HDS α ′ , α . For any fixed t ≥ 3 , given parameters 0 ≤ α ′ ≤ α < t − 1 , the input consists of degree sequences D of length n with degrees between n α ′ and 6 n α . We show that the problem is NP-complete whenever α ′ ≤ α ⁎ ( α ) : = α + ( α t − 1 − 1 ) , and is solvable in linear time when α ′ > α ⁎ ( α ) . This establishes a sharp computational phase transition separating intractable and tractable regimes, fully characterizing the complexity of sparse t -uniform hypergraphicality across all degree exponents. The result extends the earlier NP-completeness of dense hypergraphicality to a unified framework covering both sparse and dense regimes, revealing that even extremely sparse instances (with maximum degree o ( n ) but Ω ( n t − 1 t ) ) remain NP-complete. On the other hand, the t -uniform hypergraphicality problem is solvable in linear time when the maximum degree is o ( n t − 1 t ) . This dichotomy provides a comprehensive classification of the complexity landscape for hypergraphical degree sequences.

Source: Discrete Mathematics

DOI: 10.1016/j.disc.2026.115435

Authors: István Miklós, Miklós Ruszinkó, Bogdán Zavalnij

Institutions: HUN-REN Alfréd Rényi Institute of Mathematics, HUN-REN Institute for Computer Science and Control, Sorbonne University Abu Dhabi

Language: English

Topic: Advanced Graph Theory Research

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,320

Funders: European Commission, Nemzeti Kutatási Fejlesztési és Innovációs Hivatal, European Research Council