2016
DOI: 10.1002/jgt.22071
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Partite Saturation Problems

Abstract: Abstract. We look at several saturation problems in complete balanced blow-ups of graphs. We let H[n] denote the blow-up of H onto parts of size n and refer to a copy of H in H[n] as partite if it has one vertex in each part of H[n]. We then ask how few edges a subgraph G of H[n] can have such that G has no partite copy of H but such that the addition of any new edge from H[n] creates a partite H. When H is a triangle this value was determined by Ferrara, Jacobson, Pfender, and Wenger in [5]. Our main result i… Show more

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Cited by 5 publications
(4 citation statements)
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“…We remark that this phenomenon does not happen for partite saturation. Roberts [13] studied the corresponding more general problem for sat(K r×n , K r ) and showed that the minimum number of edges in a K r -saturated subgraph of K r×n where the subgraph is allowed to contain K r is r 2 (2n − 1) for r ≥ 4 and sufficiently large n. On the other hand, Theorem 1 and Theorem 2 imply that sat(K r×n , K r ) ≥ r(2r−4)n+o(n) > r 2 (2n − 1) for sufficiently large n.…”
Section: Discussionmentioning
confidence: 99%
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“…We remark that this phenomenon does not happen for partite saturation. Roberts [13] studied the corresponding more general problem for sat(K r×n , K r ) and showed that the minimum number of edges in a K r -saturated subgraph of K r×n where the subgraph is allowed to contain K r is r 2 (2n − 1) for r ≥ 4 and sufficiently large n. On the other hand, Theorem 1 and Theorem 2 imply that sat(K r×n , K r ) ≥ r(2r−4)n+o(n) > r 2 (2n − 1) for sufficiently large n.…”
Section: Discussionmentioning
confidence: 99%
“…This function was first studied recently by Ferrara, Jacobson, Pfender and Wenger [7] who determined sat(n, k, 3) for n ≥ 100. Later, Roberts [13] showed that sat(n, 4, 4) = 18n − 21 for sufficiently large n.…”
Section: Introductionmentioning
confidence: 99%
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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: 96%