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

Draganić, Nemanja; Krivelevich, Michael; Nenadov, Rajko

doi:10.1090/tran/8660

Cited by 11 publications

(12 citation statements)

References 45 publications

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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%

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Hamilton cycles in pseudorandom graphs

Glock¹,

Munhá²,

Sudakov³

2023

Preprint

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Finding general conditions which ensure that a graph is Hamiltonian is a central topic in graph theory. An old and well known conjecture in the area states that any d-regular n-vertex graph G whose second largest eigenvalue in absolute value λ(G) is at most d/C, for some universal constant C > 0, has a Hamilton cycle. In this paper, we obtain two main results which make substantial progress towards this problem. Firstly, we settle this conjecture in full when the degree d is at least a small power of n. Secondly, in the general case we show that λ(G) ≤ d/C(log n) 1/3 implies the existence of a Hamilton cycle, improving the 20-year old bound of d/ log 1−o(1) n of Krivelevich and Sudakov. We use in a novel way a variety of methods, such as a robust Pósa rotation-extension technique, the Friedman-Pippenger tree embedding with rollbacks and the absorbing method, combined with additional tools and ideas.Our results have several interesting applications, giving best bounds on the number of generators which guarantee the Hamiltonicity of random Cayley graphs, which is an important partial case of the well known Hamiltonicity conjecture of Lovász. They can also be used to improve a result of Alon and Bourgain on additive patterns in multiplicative subgroups.

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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%

Hamilton cycles in pseudorandom graphs

Glock¹,

Munhá²,

Sudakov³

2023

Preprint

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“…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%

Effective bounds for induced size-Ramsey numbers of cycles

Domagoj¹,

Draganić²,

Sudakov³

2023

Preprint

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The induced size-Ramsey number rk ind (H) of a graph H is the smallest number of edges a (host) graph G can have such that for any k-coloring of its edges, there exists a monochromatic copy of H which is an induced subgraph of G. In 1995, in their seminal paper, Haxell, Kohayakawa and Łuczak showed that for cycles, these numbers are linear for any constant number of colours, i.e., rk ind (C n ) ≤ Cn for some C = C(k). The constant C comes from the use of the regularity lemma, and has a tower type dependence on k. In this paper we significantly improve these bounds, showing that rk)n when n is even, thus obtaining only a polynomial dependence of C on k. We also prove rk ind (C n ) ≤ e O(k log k) n for odd n, which almost matches the lower bound of e Ω(k) n. Finally, we show that the ordinary (non-induced) size-Ramsey number satisfies rk (C n ) = e O(k) n for odd n. This substantially improves the best previous result of e O(k 2 ) n, and is best possible, up to the implied constant in the exponent. To achieve our results, we present a new host graph construction which, roughly speaking, reduces our task to finding a cycle of approximate given length in a graph with local sparsity.

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“…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%