Covering Graphs by Monochromatic Trees and Helly-Type Results for Hypergraphs

Bucić, Matija; Korándi, Dániel; Sudakov, Benny

doi:10.1007/s00493-020-4292-9

Cited by 3 publications

(36 citation statements)

References 30 publications

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“…Since the 2-colour case was resolved completely by Kohayakawa, Mota and Schacht, the 3-colour case arises as the natural next step. This problem was recently considered by Kohayakawa, Mendonça, Mota and Schülke [14] who proved that the threshold is at most (log n/n) 1/6 , improving the previous bound of (log n/n) 1/8 from [5].…”

Section: Introductionmentioning

confidence: 91%

“…In fact, (log n/n) 1/6 is also the hard limit of the approach of [5] and was implicitly conjectured there to be the answer. On the other hand, the best known example only shows that one needs density at least (log n/n)…”

Section: Introductionmentioning

confidence: 96%

“…Bal and DeBiasio showed that the answer is when p is somewhere between (log n/n) 1/r and (log n/n) 1/(r+1) . This question was subsequently resolved by Bucić, Korándi and Sudakov in [5] who determined that the threshold for this property is (log n/n) 1/r . It is worth mentioning here that p = (log n/n) 1/k is the threshold for any k vertices in G(n, p) to have a common neighbour.…”

Section: Introductionmentioning

confidence: 99%

“…Using our new connection result we answer their question and completely determine the threshold. Perhaps surprisingly, it turns out one can do much better than in either of [5,14] and the lower bound of (log n/n)…”

Section: Introductionmentioning

confidence: 99%

“…It is worth pointing out that, unlike for us, in order to establish their weaker upper bound for this regime, [5] needed to work directly with random graphs since their connection result does not apply here. This highlights the fact that our connection result allows one to study tc r (G(n, p)) in terms of the hypergraph problem for any choice of p.…”

Section: Introductionmentioning

confidence: 99%

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Covering random graphs with monochromatic trees

Domagoj¹,

Bucić²

2021

Preprint

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Given an r-edge-coloured complete graph Kn, how many monochromatic connected components does one need in order to cover its vertex set? This natural question is a well-known essentially equivalent formulation of the classical Ryser's conjecture which, despite a lot of attention over the last 50 years, still remains open. A number of recent papers consider a sparse random analogue of this question, asking for the minimum number of monochromatic components needed to cover the vertex set of an r-edge-coloured random graph G(n, p).Recently, Bucić, Korándi and Sudakov established a connection between this problem and a certain Hellytype local to global question for hypergraphs raised about 30 years ago by Erdős, Hajnal and Tuza. We identify a modified version of the hypergraph problem which controls the answer to the problem of covering random graphs with monochromatic components more precisely. To showcase the power of our approach, we essentially resolve the 3-colour case by showing that (log n/n) 1/4 is a threshold at which point three monochromatic components are needed to cover all vertices of a 3-edge-coloured random graph, answering a question posed by Kohayakawa, Mendonça, Mota and Schülke. Our approach also allows us to determine the answer in the general r-edge coloured instance of the problem, up to lower order terms, around the point when it first becomes bounded, answering a question of Bucić, Korándi and Sudakov.

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Section: Introductionmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Covering random graphs with monochromatic trees

Domagoj¹,

Bucić²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Covering random graphs with monochromatic trees

Domagoj

Bucić

2022

Random Struct Algorithms

Self Cite

View full text Add to dashboard Cite

Given an r-edge-colored complete graph K n , how many monochromatic connected components does one need in order to cover its vertex set? This natural question is a well-known essentially equivalent formulation of the classical Ryser's conjecture which, despite a lot of attention over the last 50 years, still remains open. A number of recent papers consider a sparse random analogue of this question, asking for the minimum number of monochromatic components needed to cover the vertex set of an r-edge-colored random graph (n, p). Recently, Bucić, Korándi, and Sudakov established a connection between this problem and a certain Helly-type local to global question for hypergraphs raised about 30 years ago by Erdős, Hajnal, and Tuza. We identify a modified version of the hypergraph problem which controls the answer to the problem of covering random graphs with monochromatic components more precisely. To showcase the power of our approach, we essentially resolve the 3-color case by showing that (log n∕n) 1∕4 is a threshold at which point three monochromatic components are needed to cover all vertices of a 3-edge-colored random graph, answering a question posed by Kohayakawa, Mendonça, Mota, and Schülke. Our approach also allows us to determine the answer in the general r-edge colored instance of the problem, up to lower order terms, around the point when it first becomes bounded, answering a question of Bucić, Korándi, and Sudakov.

show abstract

Covering random graphs by monochromatic trees

Domagoj¹

2021

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Covering Graphs by Monochromatic Trees and Helly-Type Results for Hypergraphs

Cited by 3 publications

References 30 publications

Covering random graphs with monochromatic trees

Covering random graphs with monochromatic trees

Covering random graphs with monochromatic trees

Covering random graphs by monochromatic trees

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