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On the korientability of random graphs
, 2009
"... 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 korientable, wher ..."
Abstract

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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 korientable, where k ≥ 2 is an integer, if there exists an orientation of the edges such that the maximum outdegree is at most k. Let ck = sup {c: G(n, cn) is korientable w.h.p.}. We prove that for k large enough, 1 − 2 k exp −k + 1 + e −k/4) < ck/k < 1 − exp