Stability Results for Two Classes of Hypergraphs

Contiero, Lucas de Oliveira; Hoppen, Carlos; Lefmann, Hanno; Odermann, Knut

doi:10.1137/18m1190276

Cited by 4 publications

(1 citation statement)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In addition, [ 28 ] proved that K r `1 is edge-stable with respect to K r `1. Later de Oliveira Contiero, Hoppen, Lefmann, and Odermann [ 6 ], and independently, the first author [ 20 ] improved the edge-stability result by showing that a linear dependence between δ and ε is sufficient. Pikhurko [ 34 ] refined [ 28 ] by showing that T r pn, q is also the unique H r `1-free r-graph on n vertices with the maximum number of edges for sufficiently large n. Keevash [ 16 ] observed a generalization of these results to expansions of a large class of hypergraphs F .…”

Section: Cancellative Hypergraphs and Generalized Triangles An R-graphmentioning

confidence: 99%

A unified approach to hypergraph stability

Liu,

Mubayi,

Reiher

2021

Preprint

View full text Add to dashboard Cite

We present a method which provides a unified framework for most stability theorems that have been proved in graph and hypergraph theory. Our main result reduces stability for a large class of hypergraph problems to the simpler question of checking that a hypergraph H with large minimum degree that omits the forbidden structures is vertex-extendable. This means that if v is a vertex of H and H ´v is a subgraph of the extremal configuration(s), then H is also a subgraph of the extremal configuration(s). In many cases vertex-extendability is quite easy to verify. We illustrate our approach by giving new short proofs of hypergraph stability results of Pikhurko, Hefetz-Keevash, Brandt-Irwin-Jiang, Bene Watts-Norin-Yepremyan and others. Since our method always yields minimum degree stability, which is the strongest form of stability, in some of these cases our stability results are stronger than what was known earlier. Along the way, we clarify the different notions of stability that have been previously studied.

show abstract

Section: Cancellative Hypergraphs and Generalized Triangles An R-graphmentioning

confidence: 99%

A unified approach to hypergraph stability

Liu,

Mubayi,

Reiher

2021

Preprint

View full text Add to dashboard Cite

show abstract

Uniform hypergraphs with many edge‐colorings avoiding a fixed rainbow expanded complete graph

Contiero

Hoppen

Lefmann

et al. 2021

Journal of Graph Theory

Self Cite

View full text Add to dashboard Cite

For fixed positive integers r and k, and a fixed kuniform hypergraph F , we look for n-vertex hypergraphs that admit the maximum number of r-edgecolorings with the property that there is no copy of F for which all hyperedges are assigned different colors. We consider) , the k-uniform expanded complete graph, and we prove that the Turán hypergraph, the balanced ℓ-partite k-uniform hypergraph on n vertices, is the only extremal configuration for any ≥ ℓ r r k ( , ) 0 and large n.

show abstract

New short proofs to some stability theorems

Liu

2021

European Journal of Combinatorics

View full text Add to dashboard Cite

Stability Results for Two Classes of Hypergraphs

Cited by 4 publications

References 12 publications

A unified approach to hypergraph stability

A unified approach to hypergraph stability

Uniform hypergraphs with many edge‐colorings avoiding a fixed rainbow expanded complete graph

New short proofs to some stability theorems

Contact Info

Product

Resources

About