We establish a relation between two uniform models of randomk-graphs (for constantk⩾ 3) onnlabelled vertices: ℍ(k)(n,m), the randomk-graph with exactlymedges, and ℍ(k)(n,d), the randomd-regulark-graph. By extending the switching technique of McKay and Wormald tok-graphs, we show that, for some range ofd = d(n)and a constantc> 0, ifm~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 asn→ ∞. 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 whend≫ logn(or justd⩾Clogn, ifk= 3) andd=o(n1/2).