On the -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 c k = sup { c : G ( n , c n )  is  k -orientable w.h.p. } . We prove that for k large enough, 1 − 2 k exp ( − k + 1 + e − k / 4 ) < c k / k < 1 − exp ( − 2 k ( 1 − e − 2 k ) ) , and the time c k n is a threshold for the emergence of a giant subgraph of size Θ ( n ) whose edges are more than k times its vertices. Other results are presented.