TY - JOUR

T1 - ROLLING BACKWARDS CAN MOVE YOU FORWARD

T2 - ON EMBEDDING PROBLEMS IN SPARSE EXPANDERS

AU - Draganić, Nemanja

AU - Krivelevich, Michael

AU - Nenadov, Rajko

N1 - Publisher Copyright:
© 2022 American Mathematical Society

PY - 2022/7/1

Y1 - 2022/7/1

N2 - 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].

AB - 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].

UR - http://www.scopus.com/inward/record.url?scp=85132402141&partnerID=8YFLogxK

U2 - 10.1090/tran/8660

DO - 10.1090/tran/8660

M3 - מאמר

AN - SCOPUS:85132402141

VL - 375

SP - 5195

EP - 5216

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 7

ER -