2022
DOI: 10.1090/tran/8660
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Rolling backwards can move you forward: On embedding problems in sparse expanders

Abstract: We develop a general embedding method based on the Friedman-Pippenger tree embedding technique and its algorithmic version, enhanced with a roll-back idea allowing a sequential retracing of previously performed embedding steps. We use this method to obtain the following results. We show that the size-Ramsey number of logarithmically long subdivisions of bounded degree graphs is linear in their number of vertices, settling a conjecture of Pak [Proceedings of the thirteenth annual ACMSIAM symposium on discrete … Show more

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Cited by 12 publications
(12 citation statements)
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“…This vertex-disjoint paths problem has been extensively studied in the literature. We benefit here tremendously from a recent "roll-back technique" which allows one to indeed find all these connections in expanding graphs (see [24]). Roughly speaking, this technique allows us, for a given pair of vertices we want to connect, to build two binary trees rooted at both vertices in a controlled fashion (using a concept of Friedman and Pippenger [26]); then, when these trees are large enough, we are guaranteed by the pseudorandomness of G to find an edge between them which then obviously leads to a path connecting the pair.…”
Section: Graphs With Many Cycles and Theorem 13mentioning
confidence: 99%
See 1 more Smart Citation
“…This vertex-disjoint paths problem has been extensively studied in the literature. We benefit here tremendously from a recent "roll-back technique" which allows one to indeed find all these connections in expanding graphs (see [24]). Roughly speaking, this technique allows us, for a given pair of vertices we want to connect, to build two binary trees rooted at both vertices in a controlled fashion (using a concept of Friedman and Pippenger [26]); then, when these trees are large enough, we are guaranteed by the pseudorandomness of G to find an edge between them which then obviously leads to a path connecting the pair.…”
Section: Graphs With Many Cycles and Theorem 13mentioning
confidence: 99%
“…Then, a simple but powerful observation allows one to remove leaves from this tree so that this invariant is still maintained. We refer the reader to [24] for more on the topic. First, we give the definition of this invariant -a good embedding.…”
Section: Lemma 313 ([10]mentioning
confidence: 99%
“…Beck [2] showed that this is true for paths, which was later extended to all boundeddegree trees by Friedman and Pippenger [16]. It is also known that logarithmic subdivisions of bounded degree graphs have linear size-Ramsey numbers [7], as well as bounded degree graphs with bounded treewidth [21]. Given all of the mentioned results, it might be tempting to assume that all graphs of bounded degree have linear size-Ramsey numbers.…”
Section: Introductionmentioning
confidence: 98%
“…Most of the known families with linear size-Ramsey numbers have a bounded structural parameter, such as bandwidth [5] or, more generally, treewidth [15] (though see the recent papers [8,18] for examples with a somewhat different flavour). However, a fairly simple family of graphs which does not fall into any of these categories, but which may still have linear size-Ramsey numbers, is the family of two-dimensional grid graphs.…”
Section: Introductionmentioning
confidence: 99%