Saturation in random graphs
Dániel Korándi,Benny Sudakov +1 more
TL;DR: The problem of minimizing the number of edges in a maximal Ks-free subgraph of the Erdi¾?s-Renyi random graph was studied in this paper.
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Abstract: A graph H is Ks-saturated if it is a maximal Ks-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 Ks-saturated graph was determined over 50 years ago by Zykov and independently by Erdi¾?s, Hajnal and Moon. In this paper, we study the random analog of this problem: minimizing the number of edges in a maximal Ks-free subgraph of the Erdi¾?s-Renyi random graph Gn, 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. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 169-181, 2017
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Citations
Rainbow saturation and graph capacities
TL;DR: In this article, the minimum size of a graph on n vertices that contains no rainbow copy of the original graph but the addition of any missing edge is defined. And the number of missing edges is the saturation number of the graph.
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Linearity of saturation for Berge hypergraphs
TL;DR: In this paper, it was shown that sat k ( n, Berge- F ) = O( n ) for all graphs F and uniformities 3 ≤ k ≤ 5, partially answering a conjecture of English, Gordon, Graber, Methuku, and Sullivan.
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Star saturation number of random graphs
TL;DR: In this article, the F -saturation number of the Erdős-Renyi random graph G (n, p ) for any complete graph F is determined asymptotically for star graphs.
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•Posted Content
Linearity of Saturation for Berge Hypergraphs
TL;DR: It is shown that sat k ( n, Berge- F ) = O ( n ) for all graphs F and uniformities 3 ≤ k ≤ 5 , partially answering a conjecture of English, Gordon, Graber, Methuku, and Sullivan.
Triangles in Ks-saturated graphs with minimum degree t
Craig Timmons,Benjamin Cole,Albert Curry,David Davini +3 more
- 01 Jan 2020
TL;DR: In this paper, it was shown that the minimum number of triangles in an n-vertex k-saturated graph with minimum degree 4 is exactly 2n-4 and that there is a unique extremal graph.
References
A Problem in Graph Theory
TL;DR: In this article, it is shown that with the addition of any new edge a compIete k-graph is formed, where each edge joins a vertex to itself and at most one edge joins any two vertices.
325
Independent sets in hypergraphs
TL;DR: In this article, the authors give a structural characterization of the independent sets in a large class of uniform hypergraphs by showing that every independent set is almost contained in one of a small number of relatively sparse sets and then derive many interesting results as fairly straightforward consequences of this abstract theorem.
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