Hamiltonian degree sequences in digraphs

We show that for each η > 0 every digraph G of sufficiently large order n is Hamiltonian if its out- and indegree sequences d+ 1 ≤ · · · ≤ d+ n and d− 1 ≤ · · · ≤ d− n satisfy (i) d+ i ≥ i + ηn or d− n−i−ηn ≥ n − i and (ii) d− i ≥ i + ηn or d+ n−i−ηn ≥ n − i for all i < n/2. This gives an approximate solution to a problem of Nash-Williams [22] concerning a digraph analogue of Chv´atal’s theorem. In fact, we prove the stronger result that such digraphs G are pancyclic.

• Type: Journal
• Year: 2010
• Where: JCT
• Authors: Daniela Kühn, Deryk Osthus, Andrew Treglown
• Keywords: Approximate Solution, Indegree Sequences, JCT 2010, Large Order