A Density Result For Random Sparse Oriented Graphs And Its Relation To A Conjecture Of Woodall

We prove that for all # 3and#>0 there exists a sparse oriented graph of arbitrarily large order with oriented girth # andsuchthatany1/2+# proportion of its arcs induces an oriented cycle of length #. As a corollary we get that there exist infinitely many oriented graphs with vanishing density of oriented girth # such that deleting any 1/#-fraction of their edges does not destroy all their oriented cycles. The proof is probabilistic.