The size-Ramsey number of short subdivisions

Draganić, Nemanja; Krivelevich, Michael; Nenadov, Rajko

doi:10.48550/arxiv.2004.14139

Cited by 2 publications

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“…Size-Ramsey numbers of 'short' subdivisions were first studied by Kohayakawa, Retter and Rödl [41]. In a recent paper [23] we improved their bounds by showing that Rk (H q ) ≤ O(n 1+1/q ), for constant q, k and for all bounded degree graphs H, thus removing a polylogarithmic factor from their bound and answering their question. In general, these graphs were considered in the context of Ramsey theory by Burr and Erdős [15] and by Alon [3].…”

Section: Subdivisions Of Graphsmentioning

confidence: 94%

“…Extending the definition in [42], we say that a graph G is k-partition universal for a class of graphs F if for every kcoloring of the edges of G, there exists a monochromatic subgraph of G which contains a copy of every graph in F. Under this framework, we actually prove that the graph we color is up to a constant factor the optimal k-partition universal graph for the class of all described subdivisions of graphs. For further universality-type results in Ramsey theory see for example [19,23,41,42].…”

Section: Subdivisions Of Graphsmentioning

confidence: 99%

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Rolling backwards can move you forward: on embedding problems in sparse expanders

Draganić¹,

Krivelevich²,

Nenadov³

2021

Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)

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We develop a general embedding method based on the Friedman-Pippenger tree embedding technique (1987) and its algorithmic version, essentially due to Aggarwal et al. (1996), enhanced with a roll-back idea allowing to sequentially retrace previously performed embedding steps. This proves to be a powerful tool for embedding graphs of large girth into expander graphs. As an application of this method, we settle two problems:• For a graph H, we denote by H q the graph obtained from H by subdividing its edges with q−1 vertices each. We show that the k-size-Ramsey number Rk (H q ) satisfies Rk (H q ) = O(qn) for every bounded degree graph H on n vertices and for q = Ω(log n), which is optimal up to a constant factor. This settles a conjecture of Pak ( 2002). • We give a deterministic, polynomial time algorithm for finding vertex-disjoint paths between given pairs of vertices in a strong expander graph. More precisely, let G be an (n, d, λ)-graph with λ = O(d 1−ε ), and let P be any collection of at most c n log d log n disjoint pairs of vertices in G for some small constant c, such that in the neighborhood of every vertex in G there are at most d/4 vertices from P. Then there exists a polynomial time algorithm which finds vertex-disjoint paths between every pair in P, and each path is of the same length = O log n log d . Both the number of pairs and the length of the paths are optimal up to a constant factor; the result answers the offline version of a question of Alon and Capalbo (2007).

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Section: Subdivisions Of Graphsmentioning

confidence: 94%

Section: Subdivisions Of Graphsmentioning

confidence: 99%

Rolling backwards can move you forward: on embedding problems in sparse expanders

Draganić¹,

Krivelevich²,

Nenadov³

2021

Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)

Self Cite

View full text Add to dashboard Cite

show abstract

“…Size-Ramsey numbers of 'short' subdivisions were first studied by Kohayakawa, Retter and Rödl [38]. In a recent paper [22] we improved their bounds by showing that Rk (H q ) ≤ O(n 1+1/q ), for constant q, k and for all bounded degree graphs H, thus removing a polylogarithmic factor from their bound and answering their question. In general, these graphs were considered in the context of Ramsey theory by Burr and Erdős [14] and by Alon [2].…”

Section: Subdivisions Of Graphsmentioning

confidence: 94%

“…Extending the definition in [39], we say that a graph G is k-partition universal for a class of graphs F if for every k-coloring of the edges of G, there exists a monochromatic subgraph of G which contains a copy of every graph in F. Under this framework, we actually prove that the graph we color is up to a constant factor the optimal k-partition universal graph for the class of all described subdivisions of graphs. For further universality-type results in Ramsey theory see for example [18,22,38,39].…”

Section: Subdivisions Of Graphsmentioning

confidence: 99%

Rolling backwards can move you forward: on embedding problems in sparse expanders

Draganić¹,

Krivelevich²,

Nenadov³

2020

Preprint

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We develop a general embedding method based on the Friedman-Pippenger tree embedding technique (1987) and its algorithmic version, essentially due to Aggarwal et al. (1996), enhanced with a roll-back idea allowing to sequentially retrace previously performed embedding steps. This proves to be a powerful tool for embedding graphs of large girth into expander graphs. As an application of this method, we settle two problems:• For a graph H, we denote by H q the graph obtained from H by subdividing its edges with q−1 vertices each. We show that the k-size-Ramsey number Rk (H q ) satisfies Rk (H q ) = O(qn) for every bounded degree graph H on n vertices and for q = Ω(log n), which is optimal up to a constant factor. This settles a conjecture of Pak ( 2002). • We give a deterministic, polynomial time algorithm for finding vertex-disjoint paths between given pairs of vertices in a strong expander graph. More precisely, let G be an (n, d, λ)-graph with λ = O(d 1−ε ), and let P be any collection of at most c n log d log n disjoint pairs of vertices in G for some small constant c, such that in the neighborhood of every vertex in G there are at most d/4 vertices from P. Then there exists a polynomial time algorithm which finds vertex-disjoint paths between every pair in P, and each path is of the same length ℓ = O log n log d . Both the number of pairs and the length of the paths are optimal up to a constant factor; the result answers the offline version of a question of Alon and Capalbo (2007).

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The size-Ramsey number of short subdivisions

Cited by 2 publications

References 0 publications

Rolling backwards can move you forward: on embedding problems in sparse expanders

Rolling backwards can move you forward: on embedding problems in sparse expanders

Rolling backwards can move you forward: on embedding problems in sparse expanders

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