Hamiltonian degree sequences in digraphs

Abstract

We show that for each η > 0 every digraph G of sufficiently large order n is Hamiltonian if its out-and indegree sequences d +This gives an approximate solution to a problem of Nash- Williams (1975) [22] concerning a digraph analogue of Chvátal's theorem. In fact, we prove the stronger result that such digraphs G are pancyclic.

Finding Hamilton cycles in robustly expanding digraphs

“…A non-algorithmic version of Theorem 5 was already proved in [22]. To see that the digraphs considered in Theorem 4 are robust outexpanders, we refer the reader to Lemma 11 of [22].…”

“…To see that the digraphs considered in Theorem 4 are robust outexpanders, we refer the reader to Lemma 11 of [22]. The fact that the graphs in Theorem 2(i) are robust outexpanders is proved in Lemma 12.1 of [21].…”

“…Recently, the following approximate version of Conjecture 3 for large digraphs was proved by Kühn, Osthus and Treglown [22].…”

“…(Consider for example two disjoint cliques of equal size, joined by a matching.) For this reason, we will instead use the notion of robust outexpansion (introduced in [22] for similar reasons). Given a digraph G and S ⊆ V (G), the ν-robust out-neighbourhood of S is the set…”

“…We will also use the following lemma from [22,Lemma 11]. This is the only place where, for our proof to work, we do need our digraphs to be robust outexpanders rather than just outexpanders.…”