Saturation in random graphs

Korándi, Dániel; Sudakov, Benny

doi:10.1002/rsa.20703

Cited by 23 publications

(17 citation statements)

References 24 publications

(42 reference statements)

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“…One of the most important results on graph saturation is that for any graph F with at least one edge, there is a constant C F such that sat(n, F ) ≤ C F n. This was proved by Kászonyi and Tuza in 1986 [15], and shows that saturation numbers are linear in n. Since then, the study of saturation has become an established branch of extremal graph theory. Saturation numbers of hypergraphs and of random graphs have been studied as well [4,17,21,20]. The survey paper of Faudree, Faudree, and Schmitt [10] contains many results and references.…”

Section: History and Previous Resultsmentioning

confidence: 99%

Triangles in Ks-saturated graphs with minimum degree t

Timmons¹,

Cole²,

Curry³

et al. 2020

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For n ≥ 15, we prove that the minimum number of triangles in an n-vertex K 4saturated graph with minimum degree 4 is exactly 2n − 4, and that there is a unique extremal graph. This is a triangle version of a result of Alon, Erdős, Holzman, and Krivelevich from 1996. Additionally, we show that for any s > r ≥ 3 and t ≥ 2(s−2)+1, there is a K s-saturated n-vertex graph with minimum degree t that has s−2 r−1 2 r−1 n + c s,r,t copies of K r. This shows that unlike the number of edges, the number of K r 's (r > 2) in a K s-saturated graph is not forced to grow with the minimum degree, except for possibly in lower order terms.

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Section: History and Previous Resultsmentioning

confidence: 99%

Triangles in Ks-saturated graphs with minimum degree t

Timmons¹,

Cole²,

Curry³

et al. 2020

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“…Angel and Kolesnik [3,24] investigated the special case r = 4, and obtained the sharp asymptotics p c (K 4 ) ∼ (3n log n) −1/2 . Other questions regarding graph bootstrap percolation and weak saturation were studied, e.g., in [11,18,25]. A random H-bootstrap percolation process in which the infection of a hyperedge missing from a copy of H occurs randomly, rather than deterministically, was studied in [33].…”

Section: Introductionmentioning

confidence: 99%

The threshold for stacked triangulations

Lubetzky¹,

Peled²

2021

Preprint

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A stacked triangulation of a d-simplex o = {1, . . . , d+1} (d ≥ 2) is a triangulation obtained by repeatedly subdividing a d-simplex into d + 1 new ones via a new vertex (the case d = 2 is known as an Appolonian network).We study the occurrence of such a triangulation in the Linial-Meshulam model, i.e., for which p does the random simplicial complex Y ∼ Y d (n, p) contain the faces of a stacked triangulation of the d-simplex o, with its internal vertices labeled in [n]. In the language of bootstrap percolation in hypergraphs, it pertains to the threshold for K d+1 d+2 , the (d + 1)-uniform clique on d + 2 vertices. Our main result identifies this threshold for every d ≥ 2, showing it is asymptotically (α d n) −1/d , where α d is the growth rate of the Fuss-Catalan numbers of order d. The proof hinges on a second moment argument in the supercritical regime, and on Kalai's algebraic shifting in the subcritical regime.

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“…Very recently, Sullivan and Wenger [14] studied the analogous saturation numbers for tripartite graphs within tripartite graphs and determined sat(K n1,n2,n3 , K l,l,l ) for every fixed l ≥ 1 and every n 1 , n 2 and n 3 sufficiently large. Several other host graphs have been considered, including hypercubes [4,9,12] and random graphs [11].…”

Section: Introductionmentioning

confidence: 99%

Partite Saturation of Complete Graphs

Girão¹,

Kittipassorn²,

Popielarz³

2019

SIAM J. Discrete Math.

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We study the problem of determining sat(n, k, r), the minimum number of edges in a k-partite graph G with n vertices in each part such that G is Kr-free but the addition of an edge joining any two non-adjacent vertices from different parts creates a Kr. Improving recent results of Ferrara, Jacobson, Pfender and Wenger, and generalizing a recent result of Roberts, we define a function α(k, r) such that sat(n, k, r) = α(k, r)n + o(n) as n → ∞. Moreover, we prove thatand show that the lower bound is tight for infinitely many values of r and every k ≥ 2r − 1. This allows us to prove that, for these values, sat(n, k, r) = k(2r − 4)n + O(1) as n → ∞. Along the way, we disprove a conjecture and answer a question of the first set of authors mentioned above.

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Saturation in random graphs

Cited by 23 publications

References 24 publications

Triangles in Ks-saturated graphs with minimum degree t

Triangles in Ks-saturated graphs with minimum degree t

The threshold for stacked triangulations

Partite Saturation of Complete Graphs

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