Saturated graphs with minimal number of edges

Kászonyi, László; Tuza, Źsolt

doi:10.1002/jgt.3190100209

Cited by 140 publications

(138 citation statements)

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“…Therefore for n > 2r 6 we have δ i = r − 1 for all i. Each of the x i 's has one neighbour in each adjacent part and is in a copy of K r .…”

Section: Extra-saturation Numbersmentioning

confidence: 99%

“…The study of saturation numbers was initiated by Erdős, Hajnal and Moon [3] when they proved that sat(K r , n) = (r − 2)(n − 1 2 (r − 1)). It was later shown by Kászonyi and Tuza in [6] that cliques have the largest saturation number of any graph on r vertices which in particular implies that for any H the saturation number sat(H, n) grows linearly in n.…”

Section: Introductionmentioning

confidence: 97%

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Partite Saturation Problems

Roberts

2016

Journal of Graph Theory

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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 is to calculate this value for H = K 4 when n is large. We also give exact results for paths and stars and show that for 2-connected graphs the answer is linear in n whilst for graphs which are not 2-connected the answer is quadratic in n. We also investigate a similar problem where G is permitted to contain partite copies of H but we require that the addition of any new edge from H[n] creates an extra partite copy of H. This problem turns out to be much simpler and we attain exact answers for all cliques and trees.

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“…Therefore for n > 2r 6 we have δ i = r − 1 for all i. Each of the x i 's has one neighbour in each adjacent part and is in a copy of K r .…”

Section: Extra-saturation Numbersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Partite Saturation Problems

Roberts

2016

Journal of Graph Theory

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“…The saturation number for complete graphs was determined in [2]. A systematic study by Kászonyi and Tuza [8] found the best known general upper bound for sat(n, H) in terms of the independence number of H. The saturation number is now known, often precisely, for many graphs; for these results and related problems in graph theory we refer the reader to the thorough survey of J. Faudree, R. Faudree, and Schmitt [3].…”

Section: Introductionmentioning

confidence: 99%

Saturated simple and k-simple topological graphs

Kynčl

Pach

Radoičić

et al. 2015

Computational Geometry

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A simple topological graph G is a graph drawn in the plane so that any pair of edges have at most one point in common, which is either an endpoint or a proper crossing. G is called saturated if no further edge can be added without violating this condition. We construct saturated simple topological graphs with n vertices and O(n) edges. For every k > 1, we give similar constructions for k-simple topological graphs, that is, for graphs drawn in the plane so that any two edges have at most k points in common. We show that in any k-simple topological graph, any two independent vertices can be connected by a curve that crosses each of the original edges at most 2k times. Another construction shows that the bound 2k cannot be improved. Several other related problems are also considered.

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“…This can viewed as the dual function of the celebrated Turán function ex(n, F ), the maximum number of edges in an F -saturated graph of order n. One crucial difference is that, for any fixed F , sat(n, F ) is bounded by a linear function of n (as it was shown by Kászonyi and Tuza [20]) while ex(n, F ) may be quadratic in n.…”

mentioning

confidence: 99%

“…The case of a star K 1,s is easy, see [20]. Pikhurko [25] computed the saturation function of K t * K s = K 1,...,1,s exactly when n ≥ n 0 (s, t).…”

mentioning

confidence: 99%