On the k-orientability of random graphs

Let G(n,m) be an undirected random graph with n vertices and m multiedges that may include loops, where each edge is realized by choosing its two vertices independently and uniformly at random with replacement from the set of all n vertices. The random graph G(n,m) is said to be k-orientable, where k≥2 is an integer, if there exists an orientation of the edges such that the maximum out-degree is at most k. Let ck=sup{c:G(n,cn) is k-orientable w.h.p.}. We prove that for k large enough, 1−2kexp(−k+1+e−k/4)