2022/09/27 by Zilong Yan, Yan, Zilong, Yuejian Peng +1
Computer Science · Mathematics · #Advanced Graph Theory Research #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.2209.13250
openalex publication_date 2022/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Lagrangian density of an r-uniform hypergraph H is r! multiplying the supremum of the Lagrangians of all H-free r-uniform hypergraphs. For an r-uniform graph H with t vertices, it is clear that πλ(H)≥ r!λ(Kt-1r). We say that an r-uniform hypergraph H with t vertices is λ-perfect if πλ(H)= r!λ(Kt-1r). A theorem of Motzkin and Straus implies that all 2-uniform graphs are λ-perfect. It is interesting to understand what kind of hypergraphs are λ-perfect. The property `λ-perfect' is monotone in the sense that an r-graph obtained by removing an edge from a λ-perfect r-graph (keep the same vertex set) is λ-perfect. It's interesting to understand the relation between the number of edges in a hypergraph and the `λ-perfect' property. We propose that the number of edges in a hypergraph no more than the number of edges in a linear hyperpath would guarantee the `λ-perfect' property. We show some partial result to support this conjecture. We also give some partial result to support the conjecture that the disjoint union of two λ-perfect r-uniform hypergraph is λ-perfect. We show that the disjoint union of a λ-perfect 3-graph and S2,t=\123,124,125,126,...,12(t+2)\ is perfect. This result implies the earlier result of Heftz and Keevash, Jiang, Peng and Wu, and several other earlier results.