Hamiltonian degree sequences in digraphs
arXiv:0807.1827
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^+_1\le ... \le d^+_n and d^- _1 \le ... \le d^-_n satisfy (i) d^+_i \geq i+ ηn or d^-_{n-i- ηn} \geq n-i and (ii) d^-_i \geq i+ ηn or d^+_{n-i- ηn} \geq n-i for all i < n/2. This gives an approximate solution to a problem of Nash-Williams concerning a digraph analogue of Chvátal's theorem. In fact, we prove the stronger result that such digraphs G are pancyclic.
17 pages, 2 figures. Section added which includes a proof of a conjecture of Thomassen for large tournaments. To appear in JCTB