TY - JOUR

T1 - Cycle lengths in randomly perturbed graphs

AU - Aigner-Horev, Elad

AU - Hefetz, Dan

AU - Krivelevich, Michael

N1 - Publisher Copyright:
© 2023 Wiley Periodicals LLC.

PY - 2023/12

Y1 - 2023/12

N2 - Let (Figure presented.) be an (Figure presented.) -vertex graph, where (Figure presented.) for some (Figure presented.). A result of Bohman, Frieze and Martin from 2003 asserts that if (Figure presented.), then perturbing (Figure presented.) via the addition of (Figure presented.) random edges, a.a.s. yields a Hamiltonian graph. We prove several improvements and extensions of the aforementioned result. In particular, keeping the bound on (Figure presented.) as above and allowing for (Figure presented.), we determine the correct order of magnitude of the number of random edges whose addition to (Figure presented.) a.a.s. yields a pancyclic graph. Moreover, we prove similar results for sparser graphs, and assuming the correctness of Chvátal's toughness conjecture, we handle graphs having larger independent sets. Finally, under milder conditions, we determine the correct order of magnitude of the number of random edges whose addition to (Figure presented.) a.a.s. yields a graph containing an almost spanning cycle.

AB - Let (Figure presented.) be an (Figure presented.) -vertex graph, where (Figure presented.) for some (Figure presented.). A result of Bohman, Frieze and Martin from 2003 asserts that if (Figure presented.), then perturbing (Figure presented.) via the addition of (Figure presented.) random edges, a.a.s. yields a Hamiltonian graph. We prove several improvements and extensions of the aforementioned result. In particular, keeping the bound on (Figure presented.) as above and allowing for (Figure presented.), we determine the correct order of magnitude of the number of random edges whose addition to (Figure presented.) a.a.s. yields a pancyclic graph. Moreover, we prove similar results for sparser graphs, and assuming the correctness of Chvátal's toughness conjecture, we handle graphs having larger independent sets. Finally, under milder conditions, we determine the correct order of magnitude of the number of random edges whose addition to (Figure presented.) a.a.s. yields a graph containing an almost spanning cycle.

KW - cycle lengths

KW - hamiltonicity

KW - independence number

KW - pancyclicity

KW - randomly perturbed graphs

KW - toughness

UR - http://www.scopus.com/inward/record.url?scp=85163022870&partnerID=8YFLogxK

U2 - 10.1002/rsa.21170

DO - 10.1002/rsa.21170

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AN - SCOPUS:85163022870

SN - 1042-9832

VL - 63

SP - 867

EP - 884

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

IS - 4

ER -