Weak saturation numbers of complete bipartite graphs in the clique

Kronenberg, Gal; Martins, Táısa; Morrison, Natasha

doi:10.1016/j.jcta.2020.105357

“…for 2 ≤ s ≤ t and 2 ≤ n ≤ ℓ. Moshkovitz and Shapira [15] deduced from this result that wsat(K n,n , K s,t ) = n 2 − (n − s + 1) 2 + (t − s) 2 for 2 ≤ s ≤ t ≤ n. This result can be easily generalized (see Appendix A in [11]) wsat(K n,ℓ , K s,t ) = (n + ℓ − s + 1)(s − 1) + (t − s) 2 for 2 ≤ s ≤ t, n ≤ ℓ.…”

Section: Introductionmentioning

confidence: 76%

Weak saturation stability

Kalinichenko¹,

Zhukovskii²

2021

Preprint

0

View full text Add to dashboard Cite

We study wsat(G, H), the minimum number of edges in a weakly H-saturated subgraph of G. We prove that wsat(Kn, H) is stable (remains the same after independent removal of every edge of Kn with constant probability) for all pattern graphs H such that there exists a 'local' set of edges that percolates in Kn (this is true, for example, for Ks, Ks,s and Ks,s+1). Also, we find a threshold probability for the weak K1,t-saturation stability.

show abstract

“…For q = 2, Kalai [Kal85] determined wsat(n, K r,r ). Kronenberg, Martins and Morrison [KMM21] gave recently a new proof of this result, extending it to wsat(n, K r,r−1 ) and asymptotically to all wsat(n, K s,t ). No other values wsat(n, K q r1,...,r d ) are known except for r 1 = • • • = r d = 1 when H is a clique and a handful of closely related cases, e.g., when all r i but one are 1 [Pik01b].…”

Section: Introductionmentioning

confidence: 88%

“…A minor adjustment to our proof gives that, for any rational 0 < α < 1, the quantities wsat(n, H) and wsat(K αn,(1−α)n , H), when αn ∈ Z, are of the same order of magnitude. Setting H = K t,t answers the above question of [KMM21].…”

Section: Introductionmentioning

confidence: 99%

“…For q ≥ 3 while we do not have (yet) the same knowledge of limiting constants, a similar method determines asymptotically the weak saturation number of complete d-partite d-graphs in the clique, generalizing Theorem 4 of [KMM21].…”

Section: Introductionmentioning

confidence: 99%

“…. The directed weak saturation number defined above is denoted by w(K q n , K q r ), as opposed to wsat(K q n1,...,n d , K q r1,...,r d ) in the undirected setting, a similar notation was employed in [KMM21].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Weak saturation of multipartite hypergraphs

Bulavka,

Tancer,

Tyomkyn

2021

Preprint

View full text Add to dashboard Cite

Given q-uniform hypergraphs (q-graphs) F, G and H, where G is a spanning subgraph of F , G is called weakly H-saturated in F if the edges in E(F ) \ E(G) admit an ordering e1, . . . , e k so that for all i ∈ [k] the hypergraph G ∪ {e1, . . . , ei} contains an isomorphic copy of H which in turn contains the edge ei. The weak saturation number of H in F is the smallest size of an H-weakly saturated subgraph of F . Weak saturation was introduced by Bollobás in 1968, but despite decades of study our understanding of it is still limited. The main difficulty lies in proving lower bounds on weak saturation numbers, which typically withstands combinatorial methods and requires arguments of algebraic or geometrical nature.In our main contribution in this paper we determine exactly the weak saturation number of complete multipartite q-graphs in the directed setting, for any choice of parameters. This generalizes a theorem of Alon from 1985. Our proof combines the exterior algebra approach from the works of Kalai with the use of the colorful exterior algebra motivated by the recent work of Bulavka, Goodarzi and Tancer on the colorful fractional Helly theorem. In our second contribution answering a question of Kronenberg, Martins and Morrison, we establish a link between weak saturation numbers of bipartite graphs in the clique versus in a complete bipartite host graph. In a similar fashion we asymptotically determine the weak saturation number of any complete q-partite q-graph in the clique, generalizing another result of Kronenberg et al.

show abstract

Weak Saturation of Multipartite Hypergraphs

Bulavka¹,

Tancer²,

Tyomkyn³

2023

Combinatorica

1

0

View full text Add to dashboard Cite

Given q-uniform hypergraphs (q-graphs) F, G and H, where G is a spanning subgraph of F, G is called weaklyH-saturated in F if the edges in $$E(F)\setminus E(G)$$ E ( F ) \ E ( G ) admit an ordering $$e_1,\ldots , e_k$$ e 1 , … , e k so that for all $$i\in [k]$$ i ∈ [ k ] the hypergraph $$G\cup \{e_1,\ldots ,e_i\}$$ G ∪ { e 1 , … , e i } contains an isomorphic copy of H which in turn contains the edge $$e_i$$ e i . The weak saturation number of H in F is the smallest size of an H-weakly saturated subgraph of F. Weak saturation was introduced by Bollobás in 1968, but despite decades of study our understanding of it is still limited. The main difficulty lies in proving lower bounds on weak saturation numbers, which typically withstands combinatorial methods and requires arguments of algebraic or geometrical nature. In our main contribution in this paper we determine exactly the weak saturation number of complete multipartite q-graphs in the directed setting, for any choice of parameters. This generalizes a theorem of Alon from 1985. Our proof combines the exterior algebra approach from the works of Kalai with the use of the colorful exterior algebra motivated by the recent work of Bulavka, Goodarzi and Tancer on the colorful fractional Helly theorem. In our second contribution answering a question of Kronenberg, Martins and Morrison, we establish a link between weak saturation numbers of bipartite graphs in the clique versus in a complete bipartite host graph. In a similar fashion we asymptotically determine the weak saturation number of any complete q-partite q-graph in the clique, generalizing another result of Kronenberg et al.

show abstract

Weak saturation numbers of complete bipartite graphs in the clique

Cited by 12 publications

References 17 publications

Weak saturation stability

Weak saturation stability

Weak saturation of multipartite hypergraphs

Weak Saturation of Multipartite Hypergraphs

Contact Info

Product

Resources

About