Year: 2014
Type: article
Abstract: We establish a relation between two uniform models of random k -graphs (for constant k ⩾ 3) on n labelled vertices: ℍ (k) (n,m) , the random k -graph with exactly m edges, and ℍ (k) (n,d) , the random d -regular k -graph. By extending the switching technique of McKay and Wormald to k -graphs, we show that, for some range of d = d(n) and a constant c > 0, if m ~ cnd , then one can couple ℍ (k) (n,m) and ℍ (k) (n,d) so that the latter contains the former with probability tending to one as n → ∞. In view of known results on the existence of a loose Hamilton cycle in ℍ (k) (n,m) , we conclude that ℍ (k) (n,d) contains a loose Hamilton cycle when d ≫ log n (or just d ⩾ C log n , if k = 3) and d = o ( n 1/2 ).
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2121888458