An Alternative Proof of the Linearity of the Size-Ramsey Number of Paths

Dudek, Andrzej; Prałat, Paweł

doi:10.1017/s096354831400056x

Cited by 52 publications

(63 citation statements)

References 9 publications

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“…(a) The following constants satisfy conditions (16)(17)(18) (b) The following constants also satisfy conditions (16)(17)(18) Similarly, for π 1 (G n 2m,m � ), we get:…”

Section: Lemma 10mentioning

confidence: 90%

Perfect matchings and Hamiltonian cycles in the preferential attachment model

Frieze

Pérez‐Giménez

Prałat

et al. 2018

Random Struct Algorithms

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In this paper, we study the existence of perfect matchings and Hamiltonian cycles in the preferential attachment model.In this model, vertices are added to the graph one by one, and each time a new vertex is created it establishes a connection with m random vertices selected with probabilities proportional to their current degrees. (Constant m is the only parameter of the model.) We prove that if m ≥ 1253, then asymptotically almost surely there exists a perfect matching.

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“…(a) The following constants satisfy conditions (16)(17)(18) (b) The following constants also satisfy conditions (16)(17)(18) Similarly, for π 1 (G n 2m,m � ), we get:…”

Section: Lemma 10mentioning

confidence: 90%

Perfect matchings and Hamiltonian cycles in the preferential attachment model

Frieze

Pérez‐Giménez

Prałat

et al. 2018

Random Struct Algorithms

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“…We first generalize a result explicitly stated by Letzter [16, Corollary 2.1], but independently proved implicitly by Dudek and Pra lat [5] and Pokrovskiy [19].…”

Section: Loose Cyclesmentioning

confidence: 71%

Large monochromatic components and long monochromatic cycles in random hypergraphs

Bennett

DeBiasio

Dudek

et al. 2019

European Journal of Combinatorics

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We extend results of Gyárfás and Füredi on the largest monochromatic component in r-colored complete k-uniform hypergraphs to the setting of random hypergraphs. We also study long monochromatic loose cycles in r-colored random hypergraphs. In particular, we obtain a random analog of a result of Gyárfás, Sárközy, and Szemerédi on the longest monochromatic loose cycle in 2-colored complete k-uniform hypergraphs.

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“…Pick u ∈ T 1 \ (A ∪ B). By (11, 8, (12)), all but at most two vertices of A ∪ B are connected to u, contradicting (9).…”

Section: There Exist Disjoint Setsmentioning

confidence: 94%

Monochromatic Cycle Partitions of $2$-Coloured Graphs with Minimum Degree $3n/4$

Letzter

2019

Electron. J. Combin.

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Balogh, Barát, Gerbner, Gyárfás, and Sárközy proposed the following conjecture. Let G be a graph on n vertices with minimum degree at least 3n/4. Then for every 2-edge-colouring of G, the vertex set V (G) may be partitioned into two vertex-disjoint cycles, one of each colour.We prove that this conjecture holds for n large enough, improving approximate results by the aforementioned authors and by DeBiasio and Nelsen.

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An Alternative Proof of the Linearity of the Size-Ramsey Number of Paths

Cited by 52 publications

References 9 publications

Perfect matchings and Hamiltonian cycles in the preferential attachment model

Perfect matchings and Hamiltonian cycles in the preferential attachment model

Large monochromatic components and long monochromatic cycles in random hypergraphs

Monochromatic Cycle Partitions of $2$-Coloured Graphs with Minimum Degree $3n/4$

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