TY - JOUR

T1 - Long paths and cycles in random subgraphs of H-free graphs

AU - Krivelevich, Michael

AU - Samotij, Wojciech

PY - 2014/2/13

Y1 - 2014/2/13

N2 - Let H be a given finite (possibly empty) family of connected graphs, each containing a cycle, and let G be an arbitrary finite H-free graph with minimum degree at least k. For p ∈ [0, 1], we form a p-random subgraph Gp of G by independently keeping each edge of G with probability p. Extending a classical result of Ajtai, Komlós, and Szemerédi, we prove that for every positive ε, there exists a positive δ (depending only on ε) such that the following holds: If p ≥ 1+ε/k, then with probability tending to 1 as k → ∞, the random graph Gp contains a cycle of length at least nH(δk), where nH(k) > k is the minimum number of vertices in an H-free graph of average degree at least k. Thus in particular Gp as above typically contains a cycle of length at least linear in k.

AB - Let H be a given finite (possibly empty) family of connected graphs, each containing a cycle, and let G be an arbitrary finite H-free graph with minimum degree at least k. For p ∈ [0, 1], we form a p-random subgraph Gp of G by independently keeping each edge of G with probability p. Extending a classical result of Ajtai, Komlós, and Szemerédi, we prove that for every positive ε, there exists a positive δ (depending only on ε) such that the following holds: If p ≥ 1+ε/k, then with probability tending to 1 as k → ∞, the random graph Gp contains a cycle of length at least nH(δk), where nH(k) > k is the minimum number of vertices in an H-free graph of average degree at least k. Thus in particular Gp as above typically contains a cycle of length at least linear in k.

UR - http://www.scopus.com/inward/record.url?scp=84894370054&partnerID=8YFLogxK

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:84894370054

SN - 1077-8926

VL - 21

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

IS - 1

ER -