Saturated Subgraphs of the Hypercube

Johnson, J. Robert; Pinto, Trevor

doi:10.1017/s0963548316000316

Cited by 6 publications

(7 citation statements)

References 9 publications

(16 reference statements)

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“…Minimum saturation has been studied extensively in the context of graphs [1,2,5,10,12,13,18,19,20] and hypergraphs [7,14,15,16]. Such problems are typically of the following form: for a fixed (hyper)graph H, determine the minimum size of a (hyper)graph G on n vertices which does not contain a copy of H and for which adding any edge e / ∈ G, yields a (hyper)graph which contains a copy of H. This line of research was first initiated by Zykov [21] and Erdős, Hajnal and Moon [8].…”

Section: Introductionmentioning

confidence: 99%

On Saturated $k$-Sperner Systems

Morrison

Noel

Scott

2014

Electron. J. Combin.

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Given a set X, a collection F ⊆ P(X) is said to be k-Sperner if it does not contain a chain of length k + 1 under set inclusion and it is saturated if it is maximal with respect to this property. Gerbner et al. [11] conjectured that, if |X| is sufficiently large with respect to k, then the minimum size of a saturated k-Sperner system F ⊆ P(X) is 2 k−1 . We disprove this conjecture by showing that there exists ε > 0 such that for every k and |X| ≥ n 0 (k) there exists a saturated k-Sperner system F ⊆ P(X) with cardinality at most 2 (1−ε)k .A collection F ⊆ P(X) is said to be an oversaturated k-Sperner system if, for every S ∈ P(X) \ F, F ∪ {S} contains more chains of length k + 1 than F. Gerbner et al. [11] proved that, if |X| ≥ k, then the smallest such collection contains between 2 k/2−1 and O log k k 2 k elements. We show that if |X| ≥ k 2 + k, then the lower bound is best possible, up to a polynomial factor.

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Section: Introductionmentioning

confidence: 99%

On Saturated $k$-Sperner Systems

Morrison

Noel

Scott

2014

Electron. J. Combin.

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“…We will need a standard result from Coding Theory, due to Hamming [26] (for another reference, see [30,31]). This result was also used by Johnson and Pinto [27] (and we use some ideas from [27] in our proof).…”

Section: Preliminariesmentioning

confidence: 78%

“…Johnson and Pinto [27] also study another type of saturation problem, which is often referred to as semi-saturation [18,22] (other terms have also been used: see [10,28,32,34,36]). Given graphs F and H, say that a spanning subgraph G of F is (F, H)-semi-saturated if, for every edge e ∈ E(F) \ E(G), the graph G + e contains more copies of H than G does.…”

Section: Semi-saturationmentioning

confidence: 99%

Saturation in the Hypercube and Bootstrap Percolation

Morrison¹,

Noel²,

Scott³

2016

Combinator. Probab. Comp.

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Let $Q_d$ denote the hypercube of dimension $d$. Given $d\geq m$, a spanning subgraph $G$ of $Q_d$ is said to be $(Q_d,Q_m)$-saturated if it does not contain $Q_m$ as a subgraph but adding any edge of $E(Q_d)\setminus E(G)$ creates a copy of $Q_m$ in $G$. Answering a question of Johnson and Pinto, we show that for every fixed $m\geq2$ the minimum number of edges in a $(Q_d,Q_m)$-saturated graph is $\Theta(2^d)$. We also study weak saturation, which is a form of bootstrap percolation. A spanning subgraph of $Q_d$ is said to be weakly $(Q_d,Q_m)$-saturated if the edges of $E(Q_d)\setminus E(G)$ can be added to $G$ one at a time so that each added edge creates a new copy of $Q_m$. Answering another question of Johnson and Pinto, we determine the minimum number of edges in a weakly $(Q_d,Q_m)$-saturated graph for all $d\geq m\geq1$. More generally, we determine the minimum number of edges in a subgraph of the $d$-dimensional grid $P_k^d$ which is weakly saturated with respect to `axis aligned' copies of a smaller grid $P_r^m$. We also study weak saturation of cycles in the grid.Comment: 21 pages, 2 figures. To appear in Combinatorics, Probability and Computin

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“…This conjecture was independently verified by Wessel [28] and Bollobás [5], while general K s,t -saturation in bipartite graphs was later studied in [13]. Several other host graphs have also been considered, including complete multipartite graphs [11,24] and hypercubes [7,15,23].…”

Section: Introductionmentioning

confidence: 95%

Saturation in random graphs

Korándi

Sudakov

2016

Random Struct. Alg.

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A graph H is K s -saturated if it is a maximal K s -free graph, i.e., H contains no clique on s vertices, but the addition of any missing edge creates one. The minimum number of edges in a K s -saturated graph was determined over 50 years ago by Zykov and independently by Erdős, Hajnal and Moon. In this paper, we study the random analog of this problem: minimizing the number of edges in a maximal K s -free subgraph of the Erdős-Rényi random graph G(n, p). We give asymptotically tight estimates on this minimum, and also provide exact bounds for the related notion of weak saturation in random graphs. Our results reveal some surprising behavior of these parameters.

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Saturated Subgraphs of the Hypercube

Cited by 6 publications

References 9 publications

On Saturated $k$-Sperner Systems

On Saturated $k$-Sperner Systems

Saturation in the Hypercube and Bootstrap Percolation

Saturation in random graphs

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