ROLLING BACKWARDS CAN MOVE YOU FORWARD: ON EMBEDDING PROBLEMS IN SPARSE EXPANDERS

Nemanja Draganić, Michael Krivelevich, Rajko Nenadov

Research output: Contribution to journalArticlepeer-review

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 ACM-SIAM symposium on discrete algorithms (SODA’02), 2002, pp. 321-328]. • We give a deterministic, polynomial time online algorithm for finding vertex-disjoint paths of a prescribed length between given pairs of vertices in an expander graph. Our result answers a question of Alon and Capalbo [48th annual IEEE symposium on foundations of computer science (FOCS’07), 2007, pp. 518-524]. • We show that relatively weak bounds on the spectral ratio λ/d of dregular graphs force the existence of a topological minor of Kt where t = (1 − o(1))d. We also exhibit a construction which shows that the theoretical maximum t = d + 1 cannot be attained even if λ = O(d). This answers a question of Fountoulakis, Kühn and Osthus [Random Structures Algorithms 35 (2009), pp. 444-463].

Original languageEnglish
Pages (from-to)5195-5216
Number of pages22
JournalTransactions of the American Mathematical Society
Volume375
Issue number7
DOIs
StatePublished - 1 Jul 2022

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