TY - JOUR

T1 - Cycle lengths in sparse random graphs

AU - Alon, Yahav

AU - Krivelevich, Michael

AU - Lubetzky, Eyal

N1 - Publisher Copyright:
© 2021 Wiley Periodicals LLC.

PY - 2021

Y1 - 2021

N2 - We study the set (Formula presented.) of lengths of all cycles that appear in a random d-regular graph G on n vertices for (Formula presented.) fixed, as well as in binomial random graphs on n vertices with a fixed average degree (Formula presented.). Fundamental results on the distribution of cycle counts in these models were established in the 1980s and early 1990s, with a focus on the extreme lengths: cycles of fixed length, and cycles of length linear in n. Here we derive, for a random d-regular graph, the limiting probability that (Formula presented.) simultaneously contains the entire range (Formula presented.) for (Formula presented.), as an explicit expression (Formula presented.) which goes to 1 as (Formula presented.). For the random graph (Formula presented.) with (Formula presented.), where (Formula presented.) for some absolute constant (Formula presented.), we show the analogous result for the range (Formula presented.), where (Formula presented.) is the length of a longest cycle in G. The limiting probability for (Formula presented.) coincides with (Formula presented.) from the d-regular case when c is the integer (Formula presented.). In addition, for the directed random graph (Formula presented.) we show results analogous to those on (Formula presented.), and for both models we find an interval of (Formula presented.) consecutive cycle lengths in the slightly supercritical regime (Formula presented.).

AB - We study the set (Formula presented.) of lengths of all cycles that appear in a random d-regular graph G on n vertices for (Formula presented.) fixed, as well as in binomial random graphs on n vertices with a fixed average degree (Formula presented.). Fundamental results on the distribution of cycle counts in these models were established in the 1980s and early 1990s, with a focus on the extreme lengths: cycles of fixed length, and cycles of length linear in n. Here we derive, for a random d-regular graph, the limiting probability that (Formula presented.) simultaneously contains the entire range (Formula presented.) for (Formula presented.), as an explicit expression (Formula presented.) which goes to 1 as (Formula presented.). For the random graph (Formula presented.) with (Formula presented.), where (Formula presented.) for some absolute constant (Formula presented.), we show the analogous result for the range (Formula presented.), where (Formula presented.) is the length of a longest cycle in G. The limiting probability for (Formula presented.) coincides with (Formula presented.) from the d-regular case when c is the integer (Formula presented.). In addition, for the directed random graph (Formula presented.) we show results analogous to those on (Formula presented.), and for both models we find an interval of (Formula presented.) consecutive cycle lengths in the slightly supercritical regime (Formula presented.).

KW - cycle lengths

KW - regular graphs

KW - sparse random graphs

UR - http://www.scopus.com/inward/record.url?scp=85120801076&partnerID=8YFLogxK

U2 - 10.1002/rsa.21067

DO - 10.1002/rsa.21067

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AN - SCOPUS:85120801076

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

ER -