2023/10/24 by Hyunwoo Lee, Lee, Hyunwoo
Computer Science · Engineering · #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Interconnection Networks and Systems #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2310.15909
openalex publication_date 2023/10/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Dirac's theorem determines the sharp minimum degree threshold for graphs to contain perfect matchings and Hamiltonian cycles. There have been various attempts to generalize this theorem to hypergraphs with larger uniformity by considering hypergraph matchings and Hamiltonian cycles. In this paper, we consider another natural generalization of perfect matchings, Steiner triple systems. As a Steiner triple system can be viewed as a partition of pairs of vertices, it is a natural high-dimensional analogue of a perfect matching in graphs. We prove that for sufficiently large integer n with n ≡ 1 or 3 \pmod6, any n-vertex 3-uniform hypergraph H with minimum codegree at least ((3 + √(57))/(12) + o(1) )n = (0.879... + o(1))n contains a Steiner triple system. In fact, we prove a stronger statement by considering transversal Steiner triple systems in a collection of hypergraphs. We conjecture that the number (3 + √(57))/(12) can be replaced with (3)/(4) which would provide an asymptotically tight high-dimensional generalization of Dirac's theorem.