TY - JOUR
T1 - A threshold result for loose Hamiltonicity in random regular uniform hypergraphs
AU - Altman, Daniel
AU - Greenhill, Catherine
AU - Isaev, Mikhail
AU - Ramadurai, Reshma
PY - 2020/5
Y1 - 2020/5
N2 - Let G(n,r,s) denote a uniformly random r-regular s-uniform hypergraph on n vertices, where s is a fixed constant and r=r(n) may grow with n. An ℓ-overlapping Hamilton cycle is a Hamilton cycle in which successive edges overlap in precisely ℓ vertices, and 1-overlapping Hamilton cycles are called loose Hamilton cycles. When r,s≥3 are fixed integers, we establish a threshold result for the property of containing a loose Hamilton cycle. This partially verifies a conjecture of Dudek, Frieze, Ruciński and Šileikis (2015). In this setting, we also find the asymptotic distribution of the number of loose Hamilton cycles in G(n,r,s). Finally we prove that for ℓ=2,…,s−1 and for r growing moderately as n→∞, the probability that G(n,r,s) has a ℓ-overlapping Hamilton cycle tends to zero.
AB - Let G(n,r,s) denote a uniformly random r-regular s-uniform hypergraph on n vertices, where s is a fixed constant and r=r(n) may grow with n. An ℓ-overlapping Hamilton cycle is a Hamilton cycle in which successive edges overlap in precisely ℓ vertices, and 1-overlapping Hamilton cycles are called loose Hamilton cycles. When r,s≥3 are fixed integers, we establish a threshold result for the property of containing a loose Hamilton cycle. This partially verifies a conjecture of Dudek, Frieze, Ruciński and Šileikis (2015). In this setting, we also find the asymptotic distribution of the number of loose Hamilton cycles in G(n,r,s). Finally we prove that for ℓ=2,…,s−1 and for r growing moderately as n→∞, the probability that G(n,r,s) has a ℓ-overlapping Hamilton cycle tends to zero.
KW - Hamilton cycle
KW - Hypergraph
KW - Random hypergraph
KW - Threshold
UR - http://www.scopus.com/inward/record.url?scp=85075403063&partnerID=8YFLogxK
U2 - 10.1016/j.jctb.2019.11.001
DO - 10.1016/j.jctb.2019.11.001
M3 - Article
AN - SCOPUS:85075403063
SN - 0095-8956
VL - 142
SP - 307
EP - 373
JO - Journal of Combinatorial Theory, Series B
JF - Journal of Combinatorial Theory, Series B
ER -