Estimation of the k-Orientability Threshold

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 ges 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,cn) 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 (-2k(1- e
^-2k
)),and the time c
_k
n is a threshold for the emergence of a giant subgraph of size Theta(n) whose edges are more than k times its vertices. Other results are presented.